2. MAT 150 Module 7 – Operations with Functions Lesson 3 – Inverse Functions https://upload.wikimedia.org/wikipedia/comm ons/1/11/Inverse_Function_Graph.png

3. Definition of the Inverse of a Function The inverse of a function f(x) is the function that reverses the x and y values of f(x). The inverse of f(x) is called f -1 (x). The inverse of f(x) allows us to work backwards to find the x value that produced a given y value.

4. Definition of the Inverse of a Function When we say that f -1 (x) reverses the x and y values of f(x), we mean that if the point (a, b) is on the graph of f(x), then the point (b, a) is on the graph of f -1 (x). This is true for all points in the domain of f(x).

5. Definition of the Inverse of a Function Since the inverse reverses the x and y values of f(x), it reverses the domain and range. Therefore: The domain of f(x) is the range of its inverse. The range of f(x) is the domain of its inverse.

6. Definition of the Inverse of a Function Reversing the domain and range of a function is the same as reflecting f(x) across the line y = x. Thus the graphs of f(x) and f -1 (x) will be symmetric across the line y = x.

7. Definition of the Inverse of a Function

8. Determining if two functions are inverses of each other Two functions, f(x) and g(x) are inverses of each other if the two composite functions, (f◦g)(x) and (g◦f)(x), are both equal to x. We can also determine if they are inverses by checking to see if their graphs are symmetric across the line y = x.

9. Example 1: Drawing the inverse of a function

10. Example 1: Solution - Drawing the inverse of a function

11. One-to-one functions If a function f(x) has a unique y value for every x value, we say it is one-to-one. If a function is one-to-one, then it has an inverse. To check if a function is one-to-one, we check for different x values with the same y value or use the horizontal line test. The horizontal line test says that if a horizontal line crosses the graph of f(x) more than once, then the inverse of f(x) does not exist.

12. Example 2 Which one of these functions has an inverse? A. B. C. D.

13. Example 2 - Solution Which one of these functions has an inverse? A. B.

14. Example 2 - Solution Which one of these functions has an inverse? C. D.

15. Finding the inverse of a function To find the inverse from an equation, follow these steps: 1.Replace f(x) with y if necessary. 2.Swap x and y in the equation. 3.Solve for y again. 4.Replace y with f -1 (x).

16. Example 3

17. Example 3- Solution Find the inverse of the function A. f(x) = 5x - 3

18. Example 3 - Solution Find the inverse of the function B. f(x) = x 2 + 5, for x ≥ 0

19. Example 3 - Solution

20. Application of Inverse Functions Think back to lesson 1 in this module, when we discussed cost, revenue and profit. Recall that our profit function was p(x) = 65x – 5000, where x is the number of items produced. Let’s find the inverse of this function and discuss what it means.

21. Application of Inverse Functions p(x) = 65x – 5000

22. Application of Inverse Functions What does the inverse of the profit function mean? First of all, since we switched x and y to form the inverse, the meaning of x and y is also switched. That is, x now represents profit and y represents the number of items. If the original function lets us compute the profit we will make from producing x items, the inverse does the opposite: It tells us how many items we would need to produce to make $x in profit.

23. Application of Inverse Functions Use the inverse of the profit function to compute how many items must be sold to make $10,000 in profit.

24. Application of Inverse Functions