9 Elementary Functions

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I.E.S. Izpisa BelmonteSeccin Europea

Matemticas I Seccin Europea

80FUNCTIONS.

DEFINITIONS.

A function is a rule that relates how one quantity depends on other quantities.Whenever a relationship exists between two variables (or quantities) such that forevery value of the first, there is only onecorresponding value of the second, then wesay: "The second variable is a functionof the first variable."The first variable is the independentvariable (usuallyx), and the second variable isthe dependent variable (usually y). The independent variable and the dependentvariable are real numbers.We normally write functions as: f(x) and read this as "function fofx". We can useother letters for functions. Common ones are g(x) and h(x).

Domain and range.

The domainof a function is the complete set of possible values of the independentvariable in the function. In plain English, this definition means: The domain of a1function is the set of all possiblexvalues which will make the function "work" andwill output realy-values.When finding the domain, remember:

The denominator (bottom) of a fraction cannot be zero The values under a square root sign must be positive or zero The values inside a logarithm sign must be positive Sometimes the context of a real problem determines the domain

The rangeof a function is the complete set of all possible resulting valuesof thedependent variable of a function, after we have substituted the values in the domain.In plain English, the definition means: The range of a function is the possible yvalues of a function that result when we substitute all the possiblex-values into thefunction.

y = f(x)

DOMAINX

Y

RANG

E


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81When finding the range, remember: Substitute different x-values into the expression for y to see what is

happening

Make sure you look for minimumand maximumvalues of y Drawa sketch!

GRAPH OF A FUNCTION.The graph of a function f is the set of all points in the plane of the form (a, f(a)). So,the second coordinate is the image of the first one.

Normally, the values of the independent variable (generally the x-values) areplaced on the horizontal axis, while the values of the dependentvariable (generallythe y-values) are placed on the vertical axis.Thex-value, called the abscissa, is the perpendicular distance of Pfrom the y-axis.

The y-value, called the ordinate, is the perpendicular distance of Pfrom thex-axis.The values ofxand ytogether, written as (x, y) are called the co-ordinatesof thepoint P

Vertical Line TestA set of points in the plane is the graph of a function if and only if no vertical lineintersects the graph in more than one point.


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83ELEMENTARY FUNCTIONS.

1.-LINEAR FUNCTION.The term linear function is sometimes used to mean a first-degree polynomial

function of one variable. These functions are known as "linear" because they areprecisely the functions whosegraph in the Cartesian coordinate plane is a straightline.

So, the equation of a linear function is y=mx+n, where mis the slope or gradientand nis the y-intercept

The graph of the linear function y=mx+n is a straight line that passes throughthe point (0,n), so the y-intercept, n, tells us the point where the line cuts they-axis.

m is the gradient or slope and is a measure of the slant of the line(when m ispositive the straight line is an increasing function, but when m is negative

then the straight line is a decreasing function).The slope. The different values of m determine the slant of the straight line, when

m is almost 0 then the straight line is almost horizontal and when m is ahuge number then the line is almost vertical.

Two parallel lines have the same slope. You can calculate the slope using the following formula.

2.- QUADRATIC FUNCTION.

In general, a quadratic function is apolynomial function of the form

The graph of a quadratic function isa parabola whose axis of symmetry is parallel tothe y-axis. A parabola intersects its axis of

symmetry at a point called the vertex of theparabola.The vertexof a parabola is the place where it

turns, hence, it's also called the turningpoint. The x-coordinate of the vertex is

How to draw a parabola: Firstly calculate the vertex.

Determine, if it is possible the points where the function intercepts thecoordinates axes.

0a>

0a


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84 Make a chart. Plot the points. Draw the parabola.

The meaning of the coefficients:

a: If a>0 the parabola opens upward (U). If a


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874.- PIECEWISE FUNCTIONS.

A piecewise-defined function (also called a piecewise function) is a functionwhose definition changes depending on the value of theindependent variable.

5.- ABSOLUTE VALUE OF A FUNCTION.The absolute value of a real number is the number if positive and its opposite if thenumber is negative. In general the absolute value of a function is

|| {

1y

y x

2 2 1y x x

1y

3y x

22 1 0

1 0 4

3 4

x x si x

y si x

x si x

2

1 2

x si xy

si x

( )y f x ( )y f x
http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Independent_variablehttp://en.wikipedia.org/wiki/Independent_variablehttp://en.wikipedia.org/wiki/Function_(mathematics)


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886.- INVERSELY PROPORTIONAL FUNCTIONS.

Their equation isk

yx

.Graphically they are hyperbolas whose assymptotes are the

coordinates axes. The functions

ax b

y cx d

are also hyperbolas as we can seedividing.

7.- ROOT FUNCTIONS.

Function xy is continuous and increasing (when x increases, then x

increases too), the bigger x is, the more slowly the increase is and its domain

is [0, +). Function 3 xy is continuous and increasing, its domain is

xy 3xy

12

3y

x

2 5

3

xy

x


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898.- EXPONENTIAL FUNCTIONS.Their equation is y=ax, with a a positive number different to1.

Properties:

a>1 0


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10.- TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES.

NameUsual

notation

Domain of xfor real

resultRange

Graph

sine y= sinx All real numbers [-1,1]

cosine y= cos x All real numbers [-1,1]

2y Log x

1

2

y Log x

y Lnx

xy e


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tangent y=tan x }

Name Usual notation Definition Domaint Range (radians) Graph

arcsine y= arcsinx x=sin y 1 x 1 /2 y/2

arccosine y= arccos x x=cosy 1 x 1 0 y

arctangent y= arctan x x=tany all real numbers /2


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92Composition of Functions

"Function Composition" is applying one function to the results of another:

)]([)( xfgxfg . The notation g fis read as "gcircle f", or "ground f", or "gcomposed with f", "gafter f", "gfollowing f", or "gof f".

In general, this operation is not commutative, so fg is different to gf .

Examples

Given 53)()( 2 xxgxxf then 53)())(())(( 22 xxgxfgxfg andy2)53()53())(())(( xxfxgfxgf .

Inverse function.

Given y=f(x) , the inverse function of f is another one denoted 1f such as:

xxffxxff ))(())(( 1.1 .An inverse function goes in the opposite

direction!

One function and its inverse have graphs symmetric from the line y=x.

A function has an inverse if this function is injective (there arent two or morepoints with the same image).

Solve Using AlgebraYou can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function: f(x) = 2x+3

Put "y" for "f(x)": y = 2x+3

Subtract 3 from both sides: y-3 = 2x

Divide both sides by 2: (y-3)/2 = x

Swap sides: x = (y-3)/2

Solution (put "f-1(y)" for "x") : f-1(y) = (y-3)/2

This method works well for more difficult inverses.

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9 Elementary Functions