hawkes learning systems: college algebra

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Copyright © 2011 Hawkes Learning Systems. All rights reserved.

Hawkes Learning Systems:College Algebra

Section 4.6: Inverses of Functions




Objectives

o Inverses of relations.o Inverse functions and the horizontal line test.o Finding inverse function formulas.




Inverses of Relations

Let R be a relation. The inverse of R, denoted , is the set

In other words, the inverse of a relation is the set of ordered pairs of that relation with the first and second coordinates of each exchanged.

1R

1 , | , .R b a a b R

Consider the relation

The inverse of R is

, , ,7 01 2R .

1 1, ,7 0 2,R .




Example 1: Inverses of RelationsDetermine the inverse of the relation. Then graph the relation and its inverse and determine the domain and range of both.

4, 1 , 3,2 , 0,5R

1 1,4 , 2, 3 , 5,0R

In the graph to the left, R is in blue and its inverse is in red. R consists of three ordered pairs and its inverse is simply these ordered pairs with the coordinates exchanged. Note: the domain of the relation is the range of its inverse and vice versa.

Dom 4, 3,0 , Ran 1,2,5R R

1 1Dom 1,2,5 , Ran 4, 3,0R R




Example 2: Inverses of Functions

2y x

Determine the inverse of the relation. Then graph the relation and its inverse and determine the domain and range of both.

2, |R x y y x

1 2, |R x y x y

Dom , , Ran 0,R R

1 1Dom 0, , Ran ,R R

In this problem, R is described by the given equation in x and y. The inverse relation is the set of ordered pairs in R with the coordinates exchanged, so we can describe the inverse relation by just exchanging x and y in the equation, as shown at left.




Inverses of Relations

Note:A relation and its inverse are mirror images of one another (reflections) with respect to the line Even if a relation is a function, its inverse is not necessarily a function.

Verify these two facts against the previous examples.

y x .




Inverse Functions and the Horizontal Line Test

In practice, we will only be concerned with whether or not the inverse of a function f , denoted , is itself a function. Note that has already been defined: stands for the inverse of f , where we are making use of the fact that a function is also a relation.

1f

1f 1f

Caution! does not stand for when f is a function!1f 1

f





The Horizontal Line TestLet f be a function. We say that the graph of f passes the horizontal line test if every horizontal line in the plane intersects the graph no more than once.





One-to-One FunctionsA function f is one-to-one if for every pair of distinct elements and in the domain of f, we have . This means that every element of the range of f is paired with exactly one element of the domain of f.Note: If a function is one-to-one, it will pass the horizontal line test.

1x 2x 1 2f x f x





In Example 1 you have with and , then R is one-to-one, so its inverse must be a function. But, if you notice in Example 2, the graph of is a parabola and obviously fails the horizontal line test. Thus, R is not one-to-one so its inverse is not a function.

4, 1 , 3,2 , 0,5R Dom 4, ,3,0 R Ran 1,2,5R

2y x

Tip!The inverse of a function f is also a function if and only if f is one-to-one.

1f



Copyright © 2011 Hawkes Learning Systems. All rights reserved.Example 3: Inverse Functions and the

Horizontal Line Test

Does have an inverse function?

No.

4f x x

We can see by graphing this function that it does not pass the horizontal line test, as it is an open “V” shape. By this, we know that f is not one-to-one and can conclude that it does not have an inverse function.

f x



Copyright © 2011 Hawkes Learning Systems. All rights reserved.Example 4: Inverse Functions and the

Horizontal Line Test

Does have an inverse function?

Yes.

35g x x

We know that the standard cube shape passes the horizontal line test, so g has an inverse function. We can also convince ourselves of this fact algebraically:

1 2 1 25 5x x x x

3 31 25 5x x 1 2 . g x g x

g x





Consider Example 3 again:We stated in the previous slide that because f is not one-to-one, it does not have an inverse function. However, if we restrict the domain of f explicitly by specifying that the domain is the interval , the new function, with its restricted domain, is one-to-one and has an inverse function. Let’s think about this graphically: what shape does the graph f have now that we restricted the domain? Notice that it is a diagonal line beginning at the point or, simply, the right half of the graph.

This process is called restriction of domain.

4f x x

4,

0,4 ,




Finding Inverse Function Formulas

To Find a Formula forLet f be a one-to-one function, and assume that f is defined by a formula. To find a formula for , perform the following steps:1. Replace in the definition of f with the variable y. The result is an equation in x and y that is solved for y.2. Interchange x and y in the equation.3. Solve the new equation for y.4. Replace the y in the resulting equation with .

1f

1f

1f x

f x




For example, to find the inverse function formula for the function

1. Replace with y.

Finding the Inverse Function Formulas

5 1f x x 5 1y x 5 1x y

5 1y x 5 1y x

15

y x

1 15

f x x

f x

2. Interchange x and y.

3. Solve for y.

4. Replace y with 1f x .




Finding the Inverse Function Formulas

If you noticed, finding the inverse function formula for with the defined algorithm was a

relatively long process for how simple the function is. Notice that f follows a sequence of actions: first it multiplies x by 5, then it adds 1. To obtain the inverse of f we could “undo” this process by negating these actions in the reverse order. So, we would first subtract 1 and then divide by 5:

5 1f x x

1 15

. x

f x




Example 5: Finding Inverse Function Formulas

Find the inverse of the following function.

53 1f x x

1

1 51 3f x x

We can always find the inverse function formula by using the algorithm we defined. However, this function is simple enough to easily undo the actions of f in reverse order. The application of the algorithm would be:

As you might notice, for this particular function, “undoing” the actions of f in reverse order is much simpler than applying the algorithm.

5

5

5

5

3 1

3 1

3 1

3 1

f x x

y x

x y

y x

5

5

5

11 5

3 1

3 1

1 3

1 3

y x

y x

y x

f x x




Example 6: Finding Inverse Function Formulas

Find the inverse of the following function.

13 2xg xx

1

3 2xyx

13 2yxy

3 2 1y x y

3 2 1xy x y

3 2 1xy y x

3 1 2 1y x x

2 13 1

xyx

1 2 13 1

xg xx

We will apply the algorithm to find the inverse.

Substitute y for

Interchange x and y.

Solve for y. Substitute

for y. 1g x

g x .





The graph of a relation and its inverse are mirror images of one another with respect to the lineThis is still true of functions and their inverses.Consider the function , its inverse

and the graph of both: 3( ) 1 2f x x

1

1 3( ) 2 1f x x

y x .





Note: the key characteristic of the inverse of a function is that it “undoes” the function. This means that if a function and its inverse are composed together, in either order, the resulting function has no effect on any allowable input; specifically:

For reference, observe the graph on the previous slide.

1 1f f x x x f for all ,Do m and

1 . f f x x x ffor ll Doma





Consider the function for and its inverse

(you should verify that this is the inverse

of f ).

213

xf x

1 3 1f x x

1 3 1f f x f x

23 1 1

3

x

23

3

x

33x

x

21 1

3f x

xff

213 1

3x

21 1x

1 1x x

1f

1x

Below are both of the compositions of f and :

hawkes learning systems: college algebra

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