Connexions

You are here: Home » Content » Functions and Graphs - Grade 12

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• FETMaths

This module is included inLens: Siyavula: Mathematics (Gr. 10-12)
By: Siyavula

Review Status: In Review

Click the "FETMaths" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Functions and Graphs - Grade 12

Introduction

In grades 10 and 11 you have learnt about linear functions and quadratic functions as well as the hyperbolic functions and exponential functions and many more. In grade 12 you are expected to demonstrate the ability to work with various types of functions and relations including the inverses of some functions and generate graphs of the inverse relations of functions, in particular the inverses of:

y = a x + q y = a x + q
(1)
y = a x 2 y = a x 2
(2)
y = a x ; a > 0 y = a x ; a > 0
(3)

.

Definition of a Function

A function is a relation for which there is only one value of yy corresponding to any value of xx. We sometimes write y=f(x)y=f(x), which is notation meaning 'yy is a function of xx'. This definition makes complete sense when compared to our real world examples — each person has only one height, so height is a function of people; on each day, in a specific town, there is only one average temperature.

However, some very common mathematical constructions are not functions. For example, consider the relation x2+y2=4x2+y2=4. This relation describes a circle of radius 2 centred at the origin, as in Figure 1. If we let x=0x=0, we see that y2=4y2=4 and thus either y=2y=2 or y=-2y=-2. Since there are two yy values which are possible for the same xx value, the relation x2+y2=4x2+y2=4 is not the graph a function.

There is a simple test to check if a relation is a function, by looking at its graph. This test is called the vertical line test. If it is possible to draw any vertical line (a line of constant xx) which crosses the graph of the relation more than once, then the relation is not a function. If more than one intersection point exists, then the intersections correspond to multiple values of yy for a single value of xx.

We can see this with our previous example of the circle by looking at its graph again in Figure 1.

We see that we can draw a vertical line, for example the dotted line in the drawing, which cuts the circle more than once. Therefore this is not a function.

Exercises

1. State whether each of the following equations are functions or not:
1. x+y=4x+y=4
2. y=x4y=x4
3. y=2xy=2x
4. x2+y2=4x2+y2=4
2. The table gives the average per capita income, dd, in a region of the country as a function of uu, the percentage unemployed. Write down the equation to show that income is a function of the persent unemployed.
 uu 1 2 3 4 dd 22500 22000 21500 21000

Notation used for Functions

In grade 10 you were introduced to the notation used to "name" a function. In a function y=f(x)y=f(x), yy is called the dependent variable, because the value of yy depends on what you choose as xx. We say xx is the independent variable, since we can choose xx to be any number. Similarly if g(t)=2t+1g(t)=2t+1, then tt is the independent variable and gg is the function name. If f(x)=3x-5f(x)=3x-5 and you are ask to determine f(3)f(3), then you have to work out the value for f(x)f(x) when x=3x=3. For example,

f ( x ) = 3 x - 5 f ( 3 ) = 3 ( 3 ) - 5 = 4 f ( x ) = 3 x - 5 f ( 3 ) = 3 ( 3 ) - 5 = 4
(4)

Graphs of Inverse Functions

In earlier grades, you studied various types of functions and understood the effect of various parameters in the general equation. In this section, we will consider inverse functions.

An inverse function is a function which "does the reverse" of a given function. More formally, if ff is a function with domain XX, then f-1f-1 is its inverse function if and only if for every xXxX we have:

f - 1 ( f ( x ) ) = = x f - 1 ( f ( x ) ) = = x
(5)

A simple way to think about this is that a function, say y=f(x)y=f(x), gives you a yy-value if you substitute an xx-value into f(x)f(x). The inverse function tells you tells you which xx-value was used to get a particular yy-value when you substitue the yy-value into f-1(x)f-1(x). There are some things which can complicate this for example, think about a sinsin function, there are many xx-values that give you a peak as the function oscillates. This means that the inverse of a sinsin function would be tricky to define because if you substitute the peak yy-value into it you won't know which of the xx-values was used to get the peak.

y = f ( x ) we have a function y 1 = f ( x 1 ) we substitute a specific x-value into the function to get a specific y-value consider the inverse function x = f - 1 ( y ) x = f - 1 ( y ) substituting the specific y-value into the inverse should return the specific x-value = f - 1 ( y 1 ) = x 1 y = f ( x ) we have a function y 1 = f ( x 1 ) we substitute a specific x-value into the function to get a specific y-value consider the inverse function x = f - 1 ( y ) x = f - 1 ( y ) substituting the specific y-value into the inverse should return the specific x-value = f - 1 ( y 1 ) = x 1
(6)

This works both ways, if we don't have any complications like in the case of the sinsin function, so we can write:

f - 1 ( f ( x ) ) = f ( f - 1 ( x ) ) = x f - 1 ( f ( x ) ) = f ( f - 1 ( x ) ) = x
(7)

For example, if the function x3x+2x3x+2 is given, then its inverse function is x(x-2)3x(x-2)3. This is usually written as:

f : x 3 x + 2 f - 1 : x ( x - 2 ) 3 f : x 3 x + 2 f - 1 : x ( x - 2 ) 3
(8)

The superscript "-1" is not an exponent.

If a function ff has an inverse then ff is said to be invertible.

If ff is a real-valued function, then for ff to have a valid inverse, it must pass the horizontal line test, that is a horizontal line y=ky=k placed anywhere on the graph of ff must pass through ff exactly once for all real kk.

It is possible to work around this condition, by defining a “multi-valued“ function as an inverse.

If one represents the function ff graphically in a xyxy-coordinate system, the inverse function of the equation of a straight line, f-1f-1, is the reflection of the graph of ff across the line y=xy=x.

Algebraically, one computes the inverse function of ff by solving the equation

y = f ( x ) y = f ( x )
(9)

for xx, and then exchanging yy and xx to get

y = f - 1 ( x ) y = f - 1 ( x )
(10)

Figure 2
Khan academy video on inverse functions - 1

Inverse Function of y=ax+qy=ax+q

The inverse function of y=ax+qy=ax+q is determined by solving for xx as:

y = a x + q a x = y - q x = y - q a = 1 a y - q a y = a x + q a x = y - q x = y - q a = 1 a y - q a
(11)

Therefore the inverse of y=ax+qy=ax+q is y=1ax-qay=1ax-qa.

The inverse function of a straight line is also a straight line, except for the case where the straight line is a perfectly horizontal line, in which case the inverse is undefined.

For example, the straight line equation given by y=2x-3y=2x-3 has as inverse the function, y=12x+32y=12x+32. The graphs of these functions are shown in Figure 3. It can be seen that the two graphs are reflections of each other across the line y=xy=x.

Domain and Range

We have seen that the domain of a function of the form y=ax+qy=ax+q is {x:xR}{x:xR} and the range is {y:yR}{y:yR}. Since the inverse function of a straight line is also a straight line, the inverse function will have the same domain and range as the original function.

Intercepts

The general form of the inverse function of the form y=ax+qy=ax+q is y=1ax-qay=1ax-qa.

By setting x=0x=0 we have that the yy-intercept is yint=-qayint=-qa. Similarly, by setting y=0y=0 we have that the xx-intercept is xint=qxint=q.

It is interesting to note that if f(x)=ax+qf(x)=ax+q, then f-1(x)=1ax-qaf-1(x)=1ax-qa and the yy-intercept of f(x)f(x) is the xx-intercept of f-1(x)f-1(x) and the xx-intercept of f(x)f(x) is the yy-intercept of f-1(x)f-1(x).

Exercises

1. Given f(x)=2x-3f(x)=2x-3, find f-1(x)f-1(x)
2. Consider the function f(x)=3x-7f(x)=3x-7.
1. Is the relation a function?
2. If it is a function, identify the domain and range.
3. Sketch the graph of the function f(x)=3x-1f(x)=3x-1 and its inverse on the same set of axes.
4. The inverse of a function is f-1(x)=2x-4f-1(x)=2x-4, what is the function f(x)f(x)?

Inverse Function of y=ax2y=ax2

The inverse relation, possibly a function, of y=ax2y=ax2 is determined by solving for xx as:

y = a x 2 x 2 = y a x = ± y a y = a x 2 x 2 = y a x = ± y a
(12)

We see that the inverse ”function” of y=ax2y=ax2 is not a function because it fails the vertical line test. If we draw a vertical line through the graph of f-1(x)=±xf-1(x)=±x, the line intersects the graph more than once. There has to be a restriction on the domain of a parabola for the inverse to also be a function. Consider the function f(x)=-x2+9f(x)=-x2+9. The inverse of ff can be found by witing f(y)=xf(y)=x. Then

x = - y 2 + 9 y 2 = 9 - x y = ± 9 - x x = - y 2 + 9 y 2 = 9 - x y = ± 9 - x
(13)

If we restrict the domain of f(x)f(x) to be x0x0, then 9-x9-x is a function. If the restriction on the domain of ff is x0x0 then -9-x-9-x would be a function, inverse to ff.

Figure 5
Khan academy video on inverse functions - 2

Figure 6
Khan academy video on inverse functions - 3

Exercises

1. The graph of f-1f-1 is shown. Find the equation of ff, given that the graph of ff is a parabola. (Do not simplify your answer)
2. f(x)=2x2f(x)=2x2.
1. Draw the graph of ff and state its domain and range.
2. Find f-1f-1 and, if it exists, state the domain and range.
3. What must the domain of ff be, so that f-1f-1 is a function ?
3. Sketch the graph of x=-10-y2x=-10-y2. Label a point on the graph other than the intercepts with the axes.
1. Sketch the graph of y=x2y=x2 labelling a point other than the origin on your graph.
2. Find the equation of the inverse of the above graph in the form y=...y=....
3. Now sketch the graph of y=xy=x.
4. The tangent to the graph of y=xy=x at the point A(9;3) intersects the xx-axis at B. Find the equation of this tangent and hence or otherwise prove that the yy-axis bisects the straight line AB.
4. Given: g(x)=-1+xg(x)=-1+x, find the inverse of g(x)g(x) in the form g-1(x)=...g-1(x)=....

Inverse Function of y=axy=ax

The inverse function of y=ax2y=ax2 is determined by solving for xx as follows:

y = a x log ( y ) = log ( a x ) = x log ( a ) x = log ( y ) log ( a ) y = a x log ( y ) = log ( a x ) = x log ( a ) x = log ( y ) log ( a )
(14)

The inverse of y=10xy=10x is x=10yx=10y, which we write as y=log(x)y=log(x). Therefore, if f(x)=10xf(x)=10x, then f-1=log(x)f-1=log(x).

The exponential function and the logarithmic function are inverses of each other; the graph of the one is the graph of the other, reflected in the line y=xy=x. The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

Exercises

1. Given that f(x)=(15)xf(x)=(15)x, sketch the graphs of ff and f-1f-1 on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is ff and which is f-1f-1.
2. Given that f(x)=4-xf(x)=4-x,
1. Sketch the graphs of ff and f-1f-1 on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is ff and which is f-1f-1.
2. Write f-1f-1 in the form y=...y=....
3. Given g(x)=-1+xg(x)=-1+x, find the inverse of g(x)g(x) in the form g-1(x)=...g-1(x)=...
1. Sketch the graph of y=x2y=x2, labeling a point other than the origin on your graph.
2. Find the equation of the inverse of the above graph in the form y=...y=...
3. Now, sketch y=xy=x.
4. The tangent to the graph of y=xy=x at the point A(9;3)A(9;3) intersects the xx-axis at BB. Find the equation of this tangent, and hence, or otherwise, prove that the yy-axis bisects the straight line ABAB.

End of Chapter Exercises

1. Sketch the graph of x=-10-y2x=-10-y2. Is this graph a function ? Verify your answer.
2. f(x)=1x-5f(x)=1x-5,
1. determine the yy-intercept of f(x)f(x)
2. determine xx if f(x)=-1f(x)=-1.
3. Below, you are given 3 graphs and 5 equations.
1. y=log3xy=log3x
2. y=-log3xy=-log3x
3. y=log3(-x)y=log3(-x)
4. y=3-xy=3-x
5. y=3xy=3x
Write the equation that best describes each graph.
4. Given g(x)=-1+xg(x)=-1+x, find the inverse of g(x)g(x) in the form g-1(x)=...g-1(x)=...
5. Consider the equation h(x)=3xh(x)=3x
1. Write down the inverse in the form h-1(x)=...h-1(x)=...
2. Sketch the graphs of h(x)h(x) and h-1(x)h-1(x) on the same set of axes, labelling the intercepts with the axes.
3. For which values of xx is h-1(x)h-1(x) undefined ?
1. Sketch the graph of y=2x2+1y=2x2+1, labelling a point other than the origin on your graph.
2. Find the equation of the inverse of the above graph in the form y=...y=...
3. Now, sketch y=xy=x.
4. The tangent to the graph of y=xy=x at the point A(9;3)A(9;3) intersects the xx-axis at BB. Find the equation of this tangent, and hence, or otherwise, prove that the yy-axis bisects the straight line ABAB.

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks