1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 4.6: Inverses of Functions

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Inverses of relations. o Inverse functions and the horizontal line test. o Finding inverse function formulas.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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 R is

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: 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.

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: 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 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.

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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.

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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 f is a function!

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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.

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Inverse Functions and the Horizontal Line Test One-to-One Functions A 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.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Inverse Functions and the Horizontal Line Test 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. Tip! The inverse of a function f is also a function if and only if f is one-to-one.

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: 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.

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: 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:

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Finding Inverse Function Formulas To Find a Formula for Let 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.

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. For example, to find the inverse function formula for the function 1. Replace with y. Finding the Inverse Function Formulas 2. Interchange x and y. 3. Solve for y. 4. Replace y with

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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:

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: 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.

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example: 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. Solve for y. Substitute for y.

18. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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:

19. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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.

20. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Finding Inverse Function Formulas Consider the function and its inverse (you should verify that this is the inverse of f ). Below are both of the compositions of f and :