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The set X is called the domain of the function.
For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.
Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates each element of X with a unique element of Y.
DOMAIN RANGE
X Y
f
x
x
x
y
y
Determine which of the following relations represent functions.
Not a function.
Function.
Function.
Not a function.(2,1) and (2,-9) both work.
Find the domain of the following functions:
A)
B)
C)
Square root is real only for nonnegative numbers.
Theorem Vertical Line Test
A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.
x
y
Not a function.
x
y
Function.
4
0
-4(0, -3)
(2, 3)
(4, 0)(10, 0)
(1, 0)
x
Domain of a function represents the horizontal spread of its graph & the range the vertical spread.
One-to-One FunctionOn-to Function
Inverse Function
A function f is said to be one-to-one or injective, if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain.
A function is not one-to-one if two different elements in the domain correspond to the same element in the range.
x1
x2
x3
y1
y2
y3
Domain Co domain
x1
x2
x3
y1
y3
DomainCo domain
One-to-one& on-to function
One-one but not on to function
NOT One-to-onebut on to function
Domain Co domain
ON-TO or Surjective function
A function f is said to be on to if every element of the co-domain is the image of some element of the domain. That is for all y in co-domain, there exist x in domain such that y = f(x).ON-TO ness depends on co-domain
BIJECTIVE FUNCTION
A function is said to be objective if it is both one-one and on-to
x1
x2
x3
y1
y2
y3
Domain Co domain
Use the graph to determine whether the functionis one-to-one.
Not one-to-one.
Horizontal lineCuts the graph in more than one point.
Use the graph to determine whether the function is one-to-one.
One-to-one.
.Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1.
Domain of f Range of f
Range of f 1 Domain of f 1
f 1
f
Domain of Range of
Range of Domain of
f f
f f
1
1
Theorem
The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.
f 1
2 0 2 4 6
2
2
4
6 f
f 1
y = x
(2, 0)
(0, 2)
Find the inverse of
The function is one-to-one.Interchange variables.
Solve for y.
Check.
Even and Odd Functions
Even functions are functions for which the left half of the plane looks like the mirror image of the right half of the plane. Odd functions are functions where the left half of the plane looks like the mirror image of the right half of the plane, only upside-down. Mathematically, we say that a function f(x) is even if f(x) = f(-x) and is odd if f(-x) = -f(x).
even functions odd functions
f(x) = |x| f(x) = 1/x
Some Examples
f(x) = x2 ,Even f(x) = x3 , Odd
y = cosx, Even y = sinx, Odd
Is there a function which is both even as well as odd?
Yes there is Only one function which is both even as well as odd
The function is y = f(x) = 0Let y = f(x) be one such function
Then, f(-x) = f(x) and f(-x) = -f(x)
So, f(x) = -f(x) f(x) = 0
PERIODIC FUNCTIONS
Periodic functions are functions that repeat over and over, or cycle on a specific period. This is expressed mathematically that A function f is periodic if there exists some number p>0 such that f(x) = f(x+p) for all possible values of xThe least possible value of p is called the
fundamental period of the function.
f(x) = sinx, is a periodic func with fundamental period 2π
f(x) = cosx, is also a periodic func with fundamental period 2π
y = tanx & y = cotx are periodic functions with fundamental period π
Graph of y = tanx
A property of some periodic functions that cycle within some definite range is that they have an amplitude in addition to a period. The amplitude of a periodic function is the distance between the highest point and the lowest point, divided by two. For example, sin(x) and cos(x) have amplitudes of 1.
If the period of f(x) is (a/b)π and that of g(x) is (c/d)π ,then the period of A.f(x) + B.g(x),where A and B are real numbers is(LCM of a,c)/(HCF of b,d) times π
COMBINATIONS OF PERIODIC FUNCTIONS
There are no hard and rigid rules for finding the periods of functions which are the combinations of periodic functions but the following technique may work in many cases.
For example, find the period of y = sin7x + tan(5/3)x.Period of sin7x is 2π/7 and that of tan(5/3)x is 3π/5.Hence the period of the given function is (LCM of 2,3)/(HCF of 7,5) times π that is 6 π
If the period of f(x) is p then that of a.f(x) + b is also p and that of f(ax+b) is p/|a|
For e.g, period of sin(4-3x) is 2π/3
If f(x) is periodic and g(x) is non periodic then f{g(x)} is not periodic except when g(x) is linear.
For e.g, y = sin(4-3x2) is not periodic
A constant function is periodic but has no fundamental period.
y = x – [x] is a periodic function whose fundamental period is 1
Increasing & Decreasing Functions
By behavior of a function, we mean, its Increasing & Decreasing
nature
BEHAVIOR OF FUNCTIONS
A function f(x) is said to be increasing in an interval, if for any x1, x2 belonging to this interval, x1 < x2 implies f(x1) ≤ f(x2) OR x1 >x2 implies f(x1) ≥ f(x2)
That is, if x increases then f(x) should increase and if x decreases then f(x) should decrease. The function is said to be strictly increasing if x1 < x2 implies f(x1) < f(x2) OR x1 >x2 implies f(x1) > f(x2)
0
5
10
The function y = 3x is strictly increasing
A function f(x) is said to be DECREASING in an interval, if for any x1, x2 belonging to this interval, x1 < x2 implies f(x1) ≥ f(x2) OR x1 > x2 implies f(x1) ≤ f(x2)
That is, if x increases then f(x) should decrease and if x decreases then f(x) should increase. The function is said to be strictly decreasing if x1 < x2 implies f(x1) > f(x2) OR x1 >x2 implies f(x1) < f(x2)
The function y = tanx is strictly increasing .
The function y = -[x] is decreasing but not strictly decreasing
DRAW THE GRAPH and verify.
MONOTONIC FUNCTION
A function is said to be MONOTONIC in an interval if it either increases or decrease in that interval but does not change its behavior.
The function y = tanx is monotonically increasing in its domain
The graph of y = cos x This function is NOT MONOTONIC
Library of Functions, Piecewise-Defined Functions
A linear function is a function of the form
f(x)=mx+b
The graph of a linear function is a line with a slope m and y-intercept b.
(0,b)
A constant function is a function of the form
f(x)=b
b
x
y
Identity function is a function of a form:
f(x)=x
(1,1)
(0,0)
y x 2
The square function
Cube Function
Square Root Function
Reciprocal Function
Absolute Value Function
f(x) = |x|
When functions are defined by more than one equation, they are called piece-wise defined functions.
For the following function
a) Find f(-1), f(1), f(3).
b) Find the domain.
c)Sketch the graph.
f(-1) = -1 + 3 = 2
f(1) = 3
f(3) = -3 + 3 = 0
a)
b)
c)
Polynomial Functions and Models
A polynomial function is a function of the form
f x x x( )= - +3 4 52
Polynomial. Degree 2.
Not a polynomial.
Not a polynomial.
Determine which of the following are polynomials. For those that are, state the degree.
(a)
(b)
(c)
If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or root of f(x) = 0. If r is a (real) zero of f, then
(a) r is an x-intercept of the graph of f.
(b) (x - r) is a factor of f.
Use the above to conclude that x = -1 and x = 4 are the real roots (zeroes) of f.
1 is a zero of multiplicity 2.-3 is a zero of multiplicity 1.-5 is a zero of multiplicity 5.
.
If r is a Zero of Odd Multiplicity
If r is a Zero of Even Multiplicity
Theorem
If f is a polynomial function of degree n, then f
has at most n-1 turning points.
TheoremFor large values of x, either positive or negative, the graph of the polynomial
resembles the graph of the power function.
For the polynomial
(a) Find the x- and y-intercepts of the graph of f.
(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.
(c) Find the power function that the graph of f resembles for large values of x.
(d) Determine the maximum number of turning points on the graph of f.
For the polynomial
(e) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis.
(f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.
(b) -4 is a zero of multiplicity 1. (crosses) -1 is a zero of multiplicity 2. (touches) 5 is a zero of multiplicity 1. (crosses)
(d) At most 3 turning points.
(a) The x-intercepts are -1, 5 and -4. y-intercept:
Test number: -5
f (-5) 160
Graph of f: Above x-axis
Point on graph: (-5, 160)
Test number: -2
f (-2) = -14
Graph of f: Below x-axis
Point on graph: (-2, -14)
-4 < x < -1
Test number: 0
f (0) = -20
Graph of f: Below x-axis
Point on graph: (0, -20)
-1 < x < 5
Test number: 6
f (6) = 490
Graph of f: Above x-axis
Point on graph: (6, 490)
8 6 4 2 0 2 4 6 8
300
100
100
300
500(6, 490)
(5, 0)(0, -20)
(-1, 0)
(-2, -14)(-4, 0)
(-5, 160)
Quadratic Functions
A quadratic function is a function of the form:
Properties of the Graph of a Quadratic Function
Parabola opens up if a > 0; the vertex is a minimum point.Parabola opens down if a < 0; the vertex is a maximum point.
a > 0Opens up
Vertex is lowest point
Axis of symmetry
Graphs of a quadratic function f(x) = ax2 + bx + c
a < 0Opens down
Vertex is highest point
Axis of symmetry
Steps for Graphing a Quadratic Function by Hand
• Determine the vertex.• Determine the axis of symmetry.• Determine the y-intercept, f(0).• Determine how many x-intercepts the graph has. • If there are no x-intercepts determine another
point from the y-intercept using the axis of symmetry.
• Graph.
Without graphing, locate the vertex and find the axis of symmetry of the following parabola. Does it open up or down?
Vertex:
Since -3 < 0 the parabola opens down.
Finding the vertex by completing the square:
= a(x - h)2 + k; vertex: (2, 13)
10 0 10
15
15
(0,0)
(2,4)
y x 2
10 0 10
15
15
(0,0)
(2, -12)
10 0 10
15
15
(2, 0)
(4, -12)
10 0 10
15
15
(2, 8)Vertex
f(x) = -3( x - 2 )2 + 8
(4,-4)
Determine whether the graph opens up or down.Find its vertex, axis of symmetry, y-intercept, x-intercept.
x-coordinate of vertex:
Axis of symmetry:
y-coordinate of vertex:
There are two x-intercepts:
5 0
10
10
Vertex: (-3, -13)
(-5.55, 0) (-0.45, 0)
(0, 5)
Quadratic Models
A farmer has 3000 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the most area?
x
w
The available fence represents the perimeter of the rectangle. If x is the length and w the width , then
2x + 2w = 3000
The area of a rectangle is represented by
A = xwLet us express one of the variables from the perimeter equation.
2x + 2w = 3000 x = (3000-2w)/2 x =1500 - w
Substitute this into the area equation and maximize for w.
A = (1500-w)w = -w2 + 1500wThe equation represents a parabola that opens down, so it has a maximum at its vertex point.
The vertex is w = -1500/(-2) = 750.
Thus the width should be 750 yards and the length is then
x =1500 - 750 = 750
The largest area field is the one with equal sides of length 750 yards and total area:
A = 7502=562,500 sq.y.
A projectile is fired from a cliff 400 feet above the water at an inclination of 45o to the horizontal, with a given muzzle velocity of 350ft per second. The height of the projectile above water is given by the equation below where x represents the horizontal distance of the projectile from the base of the cliff. Find the maximum height of the projectile.
To find the maximum height we need to find the coordinates of the vertex of the parabola that is represented by the above equation.
The maximum height is 1357.03 feet and the projectile reaches it at 1914.0625 feet from the base of the cliff.
Vertex (1914.0625, 1357.03 )
(0,400)
Rational Functions I
A rational function is a function of the form
Where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator is 0.
Find the domain of the following rational functions:
All real numbers except -6 and-2.
All real numbers except -4 and 4.
All real numbers.
y = L
y = R(x)
y
x
y = L
y = R(x)
y
x
Horizontal Asymptotes
x = cy
x
x = cy
x
Vertical Asymptotes
If an asymptote is neither horizontal nor vertical it is called oblique.
y
x
Theorem Locating Vertical Asymptotes
A rational function
in lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.
Vertical asymptotes: x = -1 and x = 1
No vertical asymptotes
Vertical asymptote: x = -4
Find the vertical asymptotes, if any, of the graph of each rational function.
1. If n < m, then y = 0 is a horizontal asymptote of the graph of R.
2. If n = m, then y = an / bm is a horizontal asymptote of the graph of R.
3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division.
4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division.
Consider the rational function
Horizontal asymptote: y = 0
Horizontal asymptote: y = 2/3
Find the horizontal and oblique asymptotes if any, of the graph of
Oblique asymptote: y = x + 6
Rational Functions II: Analyzing Graphs
Analyzing the Graph of a Rational Function
• Find the domain of the rational function R.• Write R in the lowest terms.• Locate the x and y intercepts. • Test for symmetry.• Locate vertical asymptotes. • Locate horizontal and oblique asymptotes.• Graph R.
Analyze the graph of:
x-intercept: -1
No symmetry
y-intercept:
In lowest terms:
Vertical asymptote: x = -3
Hole: (3, 4/3)
Horizontal asymptote: y = 2
-4
R(-4) = 6
Above x-axis
(-4, 6)
-2
R(-2) = -2
Below x-axis
(-2, -2)
1
R(1) = 1
Above x-axis
(1, 1)
8 6 4 2 0 2 4 6
10
5
5
10
(-4, 6)
(-2, -2) (-1, 0) (0, 2/3)
(1, 1) (3, 4/3)y = 2
x = - 3
Exponential Functions
An exponential function is a function of the form
where a is a positive real number (a > 0) and . The domain of f is the set of all real numbers.
3 2 1 0 1 2 3
2
4
6
(0, 1)
(1, 3)
(1, 6)
(-1, 1/3) (-1, 1/6)
Summary of the characteristics of the graph of
a >1
• The domain is all real numbers. Range is set of positive numbers.
• No x-intercepts; y-intercept is 1.
• a>1, is an increasing function and is one-to-one.
• The graph contains the points (0,1); (1,a), and (-1, 1/a).
• The graph is smooth continuous with no corners or gaps.
• The x-axis (y=0) is a horizontal asymptote as
3 2 1 0 1 2 3
2
4
6
(-1, 3)
(-1, 6)
(0, 1) (1, 1/3) (1, 1/6)
Summary of the characteristics of the graph of
0 <a <1
• The domain is all real numbers. Range is set of positive numbers.
• No x-intercepts; y-intercept is 1.
• 0<a<1, is a decreasing function and is one-to-one.
• The graph contains the points (0,1); (1,a), and (-1, 1/a).
• The graph is smooth continuous with no corners or gaps.
• The x-axis (y=0) is a horizontal asymptote as
0
5
10
(0, 1)
(1, 3)
y x3
0
5
10
(0, 1)
(-1, 3)
y x 3
0
5
10
(0, 3)
(-1, 5)
y = 2
y x 3 2
Horizontal Asymptote: y = 2
Range: { y | y >2 } or
Domain: All real numbers
The number e is defined as the number that the expression
In calculus this expression is expressed using limit notation as
3 2 1 0 1 2 3
2
4
6
y x2
y x3 y ex
Exponential Equations
Solve:
Logarithmic Functions
Change exponential expression into an equivalent logarithmic expression.
Change logarithmic expression into an equivalent exponential expression.
Domain of logarithmic function = Range of exponential function =
Range of logarithmic function = Domain of exponential function =
Range of Logarithmic and Exponential Functions
0
y a x
y xalog
(0, 1)
(1, 0)
y = x
a < 1
0
y a x
y xalog
(1, 0)
(0, 1)
a > 1
y = x
1. The x-intercept of the graph is 1. There is no y-intercept.
2. The y-axis is a vertical asymptote of the graph.
3. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1.
4. The graph is smooth and continuous, with no corners or gaps.
Properties of the Graph of a Logarithmic Function
The Natural Logarithm
0 5 10
5
5
(1, 0)
(e, 1)
0 5 10
5
5
(4, 0)
(e + 3, 1)
x = 3
The Common Logarithmic Function
(base=10)
Logarithmic Equations
Logarithmic and Exponential Equations
Check your answer!
Both terms are undefined.
Check x = 3.
Solution set {x | x = 3}.
Equation of Quadratic Type
No solution.
Solution x = 0.
Solution set {x | x =0}.
3 x = -10
3 x = 1 = 3 0
x
x = -1.780063868
[STEP FUNCTION]
The greatest integer function (or floor function or step function) will round any number down to the nearest integer. It is the greatest integer less than or equal to x and is denoted by [x].
For example, [2.001] =2, [2.998] = 2 and [-2.567] = -3
The domain of y = [x] is R and range is Z
Graph of y = [x]
Polynomial and Rational Inequalities
Steps for Solving Polynomial and Rational Inequalities Algebraically
• Write the inequality in one of the following forms:
where f(x) is written as a single quotient.
• Determine the numbers at which f(x) equals zero and also those numbers at which it is undefined.
• Use these numbers to separate the real line into intervals.
• Select a test number from each interval and evaluate f at the test number.
• If the value of f is positive, then f(x)> 0 for all numbers x in the interval.
• If the value of f is negative, then f(x)<0 for all numbers x in the interval.
If the inequality is not strict, include the solutions of f(x)=0 in the solution set, but do not include those where f is undefined.
Solve the inequality:
The inequality is in lowest terms, so we will first find where f(x)=0. And where is it undefined.
Undefined for x=-2
The real line is split into:
Pick x = - 3
f(-3) = -8
NEGATIVE
Pick x = -3/2 Pick x = 0 Pick x = 2
f(-3/2) = 5/2 f(0) = -1/2 f(2) = 3/4
POSITIVE NEGATIVE POSITIVE
The solution is all numbers x for which
or
Operations on Functions
The sum f + g is the function defined by
(f + g)(x) = f(x) + g(x)
The domain of f+g consists of numbers x that are in the domain of both f and g.
The difference f - g is the function defined by
(f - g)(x) = f(x) - g(x)
The domain of f - g consists of numbers x that are in the domain of both f and g.
The product f *g is the function defined by
(f * g)(x) = f(x) * g(x)
The domain of f *g consists of numbers x that are in the domain of both f and g.
Given two functions f and g, the composite function is defined by
xg(x)
x g(x)f(g(x))
g f
f(g)
Domain of g
Domain of f
Range of g Range of f
Range of f(g)
In general
Domain: x > 1
Symmetry; Graphing Key Equations
Symmetry
A graph is said to be
symmetric with respect to the x-axis
if for every point (x,y) on the graph,
the point (x,-y) is on the graph.
A graph is said to be
symmetric with respect to the y-axis
if for every point (x,y) on the graph,
the point (-x,y) is on the graph.
A graph is said to be
symmetric with respect to the origin
if for every point (x,y) on the graph,
the point (-x,-y) is on the graph.
Tests for Symmetry
• x-axisReplace y by -y in the equation. If an equivalent equation results, the graph is symmetric with respect to the x-axis.
• y-axis Replace x by -x in the equation. If an equivalent equation results, the graph is symmetric with respect to the y-axis.
• originReplace x by -x and y by -y in the equation. If an equivalent equation results, the graph is symmetric with respect to the origin.
Not symmetric with respect to the x-axis.
Symmetric with respect to the y-axis.
Not symmetric with respect to the origin.
Graphing Techniques; Transformations
4 2 0 2 4
2
4
6
8
10
(0, 0)(1, 1)
(2, 4)
(0, 2)
(1, 3)
(2, 6)
f x x( ) 2
g x x( ) 2 2
5 3 1 1 3 5
4
1
6
(0, 0) (1, 1)
(2, 4)
(0, -3)(1, -2)
(2, 1)
Vertical Shifts
• c>0 The graph of f(x) + c is the same as the graph of f(x) but shifted UP by c.
• For example: c = 2 then f(x) + 2 shifts f(x) up by 2.
• c<0 The graph of f(x) + c is the same as the graph of f(x) but shifted DOWN by c.
• For example: c = -3 then f(x) + (-3) = f(x) - 3 shifts f(x) down by 3.
Horizontal Shifts
If the argument x of a function f is replaced by x - h, h a real number, the graph of the new function y = f( x - h ) is the graph of f shifted horizontally left (if h < 0) or right (if h > 0).
Reflections about the x-Axis and the y-Axis
• The graph of g= - f(x) is the same as graph of f(x) but reflected about the x-axis.
• The graph of g= f(-x) is the same as graph of f(x) but reflected about the y-axis.
Compression and Stretches
• The graph of y = af(x) is obtained from the graph of y = f(x) by vertically stretching the graph if a > 1 or vertically compressing the graph if 0 < a < 1.
• The graph of y= f(ax) is obtained from the graph of y = f(x) by horizontally compressing the graph if a > 1 or horizontally stretching the graph if 0 < a < 1.
GRAPHS OF TRIGONOMETRIC FUNCTIONS
Here is the graph of y = sin x:
The function y = sin x has period 2π, because
sin (x + 2π) = sin x.
The graph of y = sin axWhen a function has this form,
y = sin ax,then the constant a indicates the number of periods in an interval of length 2π.For example, if a = 2, then, y = sin 2x -- that means there are 2 periods in an interval of length 2π.
If a = 3 , then y = sin 3x -- there are 3 periods in that interval:
Students are advised to draw the graphs of other
Trigonometric functions
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