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BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
BUSCALC LECTURE NOTESCHAPTER 1
Yvette Fajardo-Lim
Mathematics DepartmentDe La Salle University - Manila
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Functions
DefinitionA function, denoted by f , is a rule that assigns to eachobject x in a set X exactly one object f (x) in a set Y . Theelement f (x) in Y is called the image of x under f . The setX is called the domain of the function and Y its codomain.The set of assigned objects in Y is called the range of thefunction f , i.e., the range of f is the set {f (x)|x ∈ X} ⊆ Y.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Functions
ExampleConsider the equation y = 2x; this defines a function f forwhich the domain X is the set of all real numbers and therange of f is the set of all even numbers. Hence, f (x) = 2xis the image of x under f .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Functions
Since the value of the variable y in y = f (x) alwaysdepends on the choice of x , we say that y is the dependentvariable and since x is independent of y , x is called theindependent variable.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Composition of Functions
DefinitionGiven the functions f (x) and g(x),
1 The composition of f ◦ g is defined by(f ◦ g)(x) = f (g(x))
2 The composition of g ◦ f is defined by(g ◦ f )(x) = g(f (x))
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Composition of Functions
Example
Given f (x) = 2x2 − x + 3,g(x) = x + 2 and h(x) =√
x − 2
1 (f ◦ g)(x) = 2x2 + 7x + 92 (g ◦ f )(x) = 2x2 − x + 53 (f ◦ h)(x) = 2x −
√x − 2− 1
4 (h ◦ g)(x) =√
x
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Composition of Functions
Example
Given f (x) = 2x2 − x + 3,g(x) = x + 2 and h(x) =√
x − 2
1 (f ◦ g)(x) = 2x2 + 7x + 92 (g ◦ f )(x) = 2x2 − x + 53 (f ◦ h)(x) = 2x −
√x − 2− 1
4 (h ◦ g)(x) =√
x
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Composition of Functions
Example
Given f (x) = 2x2 − x + 3,g(x) = x + 2 and h(x) =√
x − 2
1 (f ◦ g)(x) = 2x2 + 7x + 92 (g ◦ f )(x) = 2x2 − x + 53 (f ◦ h)(x) = 2x −
√x − 2− 1
4 (h ◦ g)(x) =√
x
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Composition of Functions
Example
Given f (x) = 2x2 − x + 3,g(x) = x + 2 and h(x) =√
x − 2
1 (f ◦ g)(x) = 2x2 + 7x + 92 (g ◦ f )(x) = 2x2 − x + 53 (f ◦ h)(x) = 2x −
√x − 2− 1
4 (h ◦ g)(x) =√
x
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Composition of Functions
Example
Given f (x) = 2x2 − x + 3,g(x) = x + 2 and h(x) =√
x − 2
1 (f ◦ g)(x) = 2x2 + 7x + 92 (g ◦ f )(x) = 2x2 − x + 53 (f ◦ h)(x) = 2x −
√x − 2− 1
4 (h ◦ g)(x) =√
x
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Domain and Range of Functions
When defining a function, the domain must be given eitherimplicitly or explicitly. Unless otherwise specified, thedomain of the function is the set of all real numbers forwhich f (x) is defined. Most often, to determine the domainof a function, all values of x that result in division by 0 ortaking the root of a negative number are excluded.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Domain and Range of Functions
Example
The domain of f (x) =√
x + 4 is the set of all real numbers xsuch that x ≥ −4. The range of f is the set of allnonnegative real numbers.
Example
The domain of f (x) =2
x − 2is the set of all real numbers x
such that x 6= 2. The range of f is the set of all real numberssuch that y 6= 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Domain and Range of Functions
Example
The domain of f (x) =√
x + 4 is the set of all real numbers xsuch that x ≥ −4. The range of f is the set of allnonnegative real numbers.
Example
The domain of f (x) =2
x − 2is the set of all real numbers x
such that x 6= 2. The range of f is the set of all real numberssuch that y 6= 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Piecewise-defined Functions
Functions can also be defined using different rules ondisjoint subset of its domain. A function defined this way iscalled a piecewise-defined function.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Piecewise-defined Functions
Example
Given f (x) ={
2x + 8 if x < −4√x + 4 if x ≥ −4
f (−6) = −4 and f (5) = 3.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
The Graph of a Function
DefinitionThe graph of a function f is the set of all points{(x , y) : y = f (x)} on the plane.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
The Graph of a Function
Example
The graph of f (x) ={
2x + 8 if x < −32 if x ≥ −3
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
The Graph of a Function
Given the graph of a function f , we can determine thedomain and range of f .Collect all the vertical lines that will intersect the graph.
The intersection of the region and the x-axis shows that thedomain of the function is the set of real numbers −2 ≤ x < 2since the point (2,1) is not a point in the graph anymore.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
The Graph of a Function
To determine the range of the function, we consider theregion in the given figure which is the collection of all thehorizontal lines intersecting the graph. The intersection ofthe region and the y -axis gives the range of the function,−3 ≤ y < 1 .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Linear Functions
Definition
The slope of a line is defined as m =y2 − y1
x2 − x1, where
(x1, y1) and (x2, y2) are any two points on the line andx1 6= x2.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Linear Functions
Example
The slope of the line passing through the points (−1,3) and
(2,−2) is −53.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Linear Functions
DefinitionLinear functions are functions that have the formy = mx + b where m is the slope of the line and b is they-intercept. In the special case m = 0 with f (x) = b, we callthese functions constant functions with a horizontal line asits graph.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Slope-Intercept Form
DefinitionA linear equation in the form y = mx + b is called theslope-intercept form.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Slope-Intercept Form
ExampleThe equation of the line with slope −3 and y-intercept 2 isgiven as y = −3x + 2.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Point-Slope Form
If we know that a line passes through the point (x1, y1) andhas a slope of m then the point-slope form of the equationof the line is y − y1 = m(x − x1) .
Example
The equation of the line that passes through the point (2,3)
with slope32
is y =32
x.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Point-Slope Form
If we know that a line passes through the point (x1, y1) andhas a slope of m then the point-slope form of the equationof the line is y − y1 = m(x − x1) .
Example
The equation of the line that passes through the point (2,3)
with slope32
is y =32
x.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Two-Point Form
The line through the points (x1, y1) and (x2, y2) is given by
the two-point form y − y1 =y2 − y1
x2 − x1(x − x1)
Example
The equation of the line passing through the points (−1,3)
and (2,−2) is y =− 5x + 4
3
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
General Form
Equations of the line can be reduced to the first degreeequation in the variables x and y of the formax + by + c = 0 where a,b, and c are real numbers and aand b are not both zero. This is called the general form ofthe equation of a line.
ExampleThe equation y = −3x + 2, can be written in the generalform 3x + y − 2 = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Intercept Form
The intercept form of the equation of a line is given byxa+
yb= 1.
ExampleThe equation 8x − 3y + 24 = 0 in its intercept form is
written asx3+
y8= 1.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Intercept Form
The intercept form of the equation of a line is given byxa+
yb= 1.
ExampleThe equation 8x − 3y + 24 = 0 in its intercept form is
written asx3+
y8= 1.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Parallel and Perpendicular Lines
Equations of lines can also be determined given theequation of a parallel line or a perpendicular line. Two linesare parallel if they have the same slope and two lines areperpendicular when the product of their slopes is −1, thatis, their slopes are negative reciprocals.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Parallel and Perpendicular Lines
Example
The equation of the line that passes through (−2,3) and isparallel to 4x − 3y = 2 is 4x − 3y + 17 = 0
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Parallel and Perpendicular Lines
Example
The equation of the line that passes through (1,−4) and is
perpendicular to y = −x2+ 4 is 2x − y − 6 = 0
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Quadratic Functions
Definition
Functions of the form f (x) = ax2 + bx + c where a,b, c arereal numbers with a 6= 0 are called quadratic functions.The graph of a quadratic function is a parabola.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Quadratic Functions
Below is the graph of y = x2
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Quadratic Functions
The graph of y = x2 shows that the function has a lowestpoint at (0,0). This lowest point is called the vertex. Thevertex is the lowest point if a > 0 and the parabola opensupward; if a < 0, the vertex is the highest point and theparabola opens downward.
The vertex is at point
(−
b2a
, f
(−
b2a
))while the axis of
symmetry is the line x = −b2a
.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Quadratic Functions
The following table serves as an aid in sketching the graphof the quadratic function f (x) = ax2 + bx + c.a > 0 parabola opens upwarda < 0 parabola opens downward
vertex
(−
b2a
, f
(−
b2a
))b2 − 4ac > 0 parabola has two x-interceptsb2 − 4ac = 0 parabola has one x-interceptb2 − 4ac < 0 parabola has no x-interceptx-intercepts solutions of 0 = ax2 + bx + cy -intercept c
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Quadratic Functions
Example
Given the function f (x) = 2x2 − x − 2.a = 2 > 0 parabola opens upward
vertex
(14,−
178
)b2 − 4ac = 17 > 0 parabola has two x-intercepts
x-intercepts1±√
174
y-intercept −2
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Quadratic Functions
Graph of the function f (x) = 2x2 − x − 2.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Standard Form of Quadratic Functions
Quadratic functions may also be expressed in the formf (x) = a(x − h)2 + k ,a 6= 0. This is referred to as standardform with vertex at the point (h, k).
Example
The quadratic function f (x) = 2x2 − x − 2 in standard form
is written f (x) = 2
(x −
14
)2
−178
.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Domain and Range of Quadratic Functions
The domain of a quadratic function is the set of realnumbers. If the parabola opens upward the range of thefunction is y ≥ k and if it is downward, the range is y ≤ k .
Example
The domain of the quadratic function f (x) = 2x2 − x − 2 is
the set of real numbers while its range is y ≥ −178
.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Vertical Shift
Given the function f (x), the graph of y = f (x) + k can beobtained by shifting the graph of f (x), k units up if k > 0while if k < 0, the graph is shifted k units down.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Horizontal Shift
Consider f (x − h) when y = f (x) . If h > 0, the graphundergoes a horizontal shift h units to the right; if h < 0, thegraph undergoes a horizontal shift h units to the left.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Rational Functions
DefinitionBy a rational function f (x) we mean a function whose
assignment rule is of the form f (x) =p(x)q(x)
, where p(x) and
q(x) are polynomials and q(x) 6= 0 .
The domain of the rational functions is the set of realnumbers except for the values which will make q(x) = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Rational Functions
To graph rational functions the asymptotes must bedetermined. The vertical asymptotes occur at the domainrestriction. If p(x) = anxn + . . .+ a2x2 + a1x + a0 andq(x) = bmxm + . . .+ b2x2 + b1x + b0, the following table is asummary to aid in sketching the graph.q(r) = 0, r ∈ R vertical asymptote is the line x = rn < m horizontal asymptote is the line y = 0
n = m horizontal asymptote is the line y =an
bmn > m no horizontal asymptote
n = m + 1 oblique asymptote is the line y =p(x)q(x)
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Rational Functions
Example
Given the function f (x) =2x + 1x − 3
. The domain is the set of
all real numbers such that x 6= 3 . Thus, the verticalasymptote is x = 3 and since both the numerator anddenominator are linear, the horizontal asymptote is the liney = 2 .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Rational Functions
f (x) =2x + 1x − 3
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Rational Functions
f (x) =x2 − 4x − 3
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Inverse Functions
DefinitionA function f is said to be one-to-one if every number in itsrange corresponds to exactly one number in its domain, thatis for all x1 and x2 in the domain of f , if x1 6= x2 thenf (x1) 6= f (x2) . Equivalently, f (x1) = f (x2) only when x1 = x2.
Showing that a function is one-to-one is often a tedious andoften difficult. However, if the value of f (x) increases as thevalue of x increases for all x in its domain, then the functionis one-to-one. Similarly, if the value of f (x) decreases as thevalue of x increases for all x in its domain, the function isone-to-one.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Inverse Functions
Definition
If f is a one-to-one function then there is function f−1 , calledthe inverse of f , where f−1(f (x)) = x and f (f−1(x)) = x forall values of x in their respective domains. The domain off−1 is the range of f and the range of f−1 is the domain of f .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Inverse Functions
Example
The function f (x) = 3x − 5 is a linear function with a positiveslope and is an increasing function. Hence, its inverse
exists and f−1(x) =x + 5
3.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Inverse Functions
In general, the inverse of quadratic functions does not exist.However, if the domain will be restricted in such a way thatthe function is one-to-one on the restricted interval of thedomain, the function will have its inverse.
Example
The inverse of the function f (x) = x2 − 5, x ≤ 0 isf−1(x) = −
√x + 5.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Graph of Inverse Functions
The graph of an inverse is the reflection of the original graphabout the line y = x .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Exponential Functions
DefinitionThe exponential function with base a is defined for all realnumbers x by f (x) = ax , where a > 0, and a 6= 1 .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Exponential Functions
Example
The function f (x) = 2x is an exponential function with thegiven graph below.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Exponential Functions
The graph of the function f (x) =
(12
)x
is shown below.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Exponential Functions
1 The domain of f (x) = ax is the set of real numbers.2 The range of f (x) = ax is the set of positive real
numbers.3 The y -intercept of the graph of f (x) = ax is at point
(0,1), that is f (0) = 1.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if ax = ay then x = y .7 The graph of f (x) has a horizontal asymptote y = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Natural Exponential Function
The function f (x) = ex is called the natural exponentialfunction, where e ≈ 2.718281828. The graph of f (x) = ex
and f (x) = e−x is shown below.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Outline
1 FUNCTIONS AND THEIR GRAPHSDefinitions and ExamplesLinear FunctionsQuadratic FunctionsRational FunctionsInverse FunctionsExponential FunctionsLogarithmic Functions
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YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Logarithmic Functions
DefinitionLet a > 0 and a 6= 1. The logarithmic function with base aand written as loga , is defined by y = loga x if and only ifx = ay for every x > 0 and every real number y.
Example
log5 25 = 2 and log4164
= −3
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Logarithmic Functions
The logarithmic function with base 10 is called the commonlogarithm and we commonly write this as log x , that is, ifthe base is omitted, it is understood to be 10. Anotherspecial logarithmic function is the natural logarithm withbase e written as ln x .
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Logarithmic Functions
The functions ax and loga x are inverse functions. Hence,the graph of loga x is simply reflecting the graph of ax aboutthe line y = x .
The graph below shows 2x and log2 x
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.
BUSCALC
YvetteFajardo-Lim
FUNCTIONSAND THEIRGRAPHSDefinitions andExamples
Linear Functions
QuadraticFunctions
Rational Functions
Inverse Functions
ExponentialFunctions
LogarithmicFunctions
Properties of Logarithmic Functions
1 The domain of f (x) = loga x is the set of positive realnumbers.
2 The range of f (x) = loga x is the set of real numbers.3 The x-intercept of the graph of f (x) = loga x is at point
(1,0), that is f (1) = 0.4 If 0 < a < 1, as the value of x increases, the value of y
decreases.5 If a > 1, as the value of x increases, the value of y
increases.6 f (x) is one-to-one, if loga x = loga y then x = y .7 The graph of f (x) has a vertical asymptote x = 0.