Chapter 5 Exponential and Logarithmic Functions

1. Chapter 5 Exponential and Logarithmic Functions

2. Inverse Functions and Their Representations 5.2 Calculate inverse operations Identify one-to-one functions Find inverse functions symbolically Use other representations to find inverse functions

3. For each of the following, state the inverse actions or operations. (a) Put on a coat and go outside. (b)Subtract 7 from x and divide the result by 2. Solution Reverse order, apply inverse actions (a) Come inside and take off coat. (b) Multiply x by 2 and add 7. Example: Finding inverse actions and operations

4. Inverse Functions Gallons to Pints f(x) = 8x • Pints to Gallons • Function g performs the inverse operation of f. We say that fand g are inverse functions and write this as g(x)=f–1(x). We read f –1 as “f inverse.”

5. Inverse Functions In general, if f(a) = b , then f –1(b) = a. That is, if foutputs b with input a, then f –1 must output a with input b.Inputs and outputs (domains and ranges) are interchanged for inverse functions.

6. Composition with Inverse Functions The composition of a function with its inverse using input x produces output x.

7. One-to-One Function A function f is a one-to-one function if, for elements c and d in the domain of f, c ≠ d implies f(c) ≠ f(d). That is, different inputs always result in different outputs.

8. One-to-One and Inverse Functions If different inputs of a function fproduce the same output, then an inverse function off does notexist. However, if different inputs always produce different outputs, fis a one- to-one function. A function fis one-to-one if equal outputs always have the same input. This statement can be written as f(c) = f(d) implies c = d. Every one-to-one function has an inverse function.

9. Horizontal Line Test If every horizontal line intersects the graph of a function fat most once, then fis a one-to-one function.

10. Increasing, Decreasing andOne-to-One Functions If a continuous function f is increasing on its domain, then every horizontal line will intersect the graph of f at most once. By the horizontal line test, f is a one-to-one function. Similarly, if a continuous function g is only decreasing on its domain, then g is a one-to-one function.

11. Inverse Function Let f be a one-to-one function. Then f –1 is the inverse function of f if for every x in the domain of f and for every x in the domain of f–1.

12. Let f be a one-to-one function given by f(x) = x3 – 2. (a) Find a formula for f–1(x). (b) Identify the domain and range of f–1. (c) Verify that your result from part (a) is correct. Example: Finding and verifying an inverse function

13. Let f(x) = x3 – 2. Solution (a) Solve y = f(x) for x. Interchange x and y to obtain Replace y with f–1(x): Example: Finding and verifying an inverse function

14. Let f(x) = x3 – 2. Solution (b) Both the domain and the range of the cube root function include all real numbers. The graph of is the graph of the cube root function shifted left 2 units. Therefore the domain and range of f–1 also include all real numbers. Example: Finding and verifying an inverse function

15. Solution (c) To verify that is indeed the inverse of f(x) = x3 – 2, we must show that and that Example: Finding and verifying an inverse function

16. Solution (c) and now Example: Finding and verifying an inverse function

17. Finding a Symbolic Representation for f–1 To find a formula for f –1, perform the following steps. STEP 1: Verify that f is a one-to-one function. STEP 2: Solve the equation y = f(x) for x, obtaining the equation x = f -1(y). STEP 3: Interchange x and y to obtain y = f -1(x). To verify f –1(x), show that and

18. Numerical Representation of Inverse In the top table, fcomputes the percentage of the U.S. population with 4 or more years of college in year x.The bottom table shows a numerical representation of f–1.

19. Numerical Representation of Inverse The domains and ranges are interchanged.

20. Domains and Ranges of Inverse Functions The domain of fequals the range of f–1. The range of f equals the domain of f –1.

21. Graphs of Functions and Their Inverses The graph of f–1 is a reflection of the graph of f across the line y = x.

22. Graphs of Functions and Their Inverses If the point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f –1.

23. Let f(x)= x3+ 2. Graph f–1. Then sketch a graph of f–1. Solution Example: Representing an inverse function graphically Reflect across y = x. Calculator display.