1. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS121 Calculus I Section 0.4 Inverse Functions

2. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 0.4.1 (p. 39) Definition of Inverse Functions

3. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Example Show that and are inverses.

4. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. A Procedure for Finding the Inverse of a Function f (p. 40) How to Determine Inverse Functions

5. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Find a formula for if Example

6. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Domain of f -1 (x) = Range of f(x) Range of f -1 (x) = Domain of f(x) Why? If x is in the domain of f then y = f(x) is in the range. But, x = f -1 (y), so y is in the domain of f -1 and x is in the range. Domain and Range

7. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 0.4.9 (p. 43) Reflection Property of Inverses

8. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 0.4.5 (p. 43) Reflection Property of Inverses

9. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Existence of Inverses Suppose f(x) has an inverse. Then, for any two different x’s, say x 1 ≠ x 2,we must have f(x 1 ) ≠ f(x 2 ). Why?? Definition: We say that f(x) is one-to-one if f(x 1 ) ≠ f(x 2 ) whenever x 1 ≠ x 2.

10. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 0.4.3 (p. 42) Existence of Inverses: One-to-one

11. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Existence of Inverses Theorem: A function f(x) has an inverse if and only if it is one-to-one. Example: Which of the following are One- to-one (and thus invertible). f(x) = x 2 f(x) = x 3 + 2 f(x) = sin(x)

12. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 0.4.4 (p.42) The Horizontal Line Test Existence of Inverses

13. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. More Practice