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### Hawkes Learning Systems:College Algebra

Section 4.6: Inverses of Functions

Objectives

- Inverses of relations.
- Inverse functions and the horizontal line test.
- 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.

Consider the relation

The inverse of Ris

Example 1: Inverses of Relations

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

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.

Example 2: Inverses of Functions

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

In this problem, R is described by the given equation in xand 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.

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.

Caution!

does not stand for when fis a function!

Inverse Functions and the Horizontal Line Test

The Horizontal Line Test

Let 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.

Inverse Functions and the Horizontal Line Test

One-to-One Functions

A function f is one-to-one if for everypair 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.

Inverse Functions and the Horizontal Line Test

In Example 1you have with and , then Ris 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, Ris not one-to-one so its inverse is not a function.

Tip!

The inverse of a function fis also a function if and

only if fis one-to-one.

Example 3: Inverse Functions and the Horizontal Line Test

Does have an inverse function?

No.

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.

Example 4: Inverse Functions and the Horizontal Line Test

Does have an inverse function?

Yes.

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:

Inverse Functions and the Horizontal Line Test

Consider Example 3 again:

We stated in the previous slide that because fis not one-to-one, it does not have an inverse function. However, if we restrict the domain of fexplicitly 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.

Finding Inverse Function Formulas

To Find a Formula for

Let fbe 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 .

Finding the Inverse Function Formulas

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

1. Replace with y.

2. Interchange x and y.

3. Solve for y.

4. Replace y with

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 ffollows a sequence of actions: first it multipliesxby 5, then it adds1. To obtain the inverse of fwe could “undo” this process by negating these actions in the reverse order. So, we would first subtract 1and then divide by 5:

Example 5: Finding Inverse Function Formulas

Find the inverse of the following function.

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.

Example 6: Finding Inverse Function Formulas

Find the inverse of the following function.

We will apply the algorithm to find the inverse.

Substitute y for

Interchange x and y.

Substitute

for y.

Solve for y.

Finding Inverse Function Formulas

The graph of a relation and its inverse are mirror images of one another with respect to the line

This is still true of functions and their inverses.

Consider the function , its inverse

and the graph of both:

Finding Inverse Function Formulas

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.

Finding Inverse Function Formulas

Consider the function for and its inverse (you should verify that this is the inverse of f ).

Below are both of the compositions of fand :

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