1. Inverse Functions MATH 109 - Precalculus S. Rook

2. Overview Section 1.9 in the textbook: – One-to-one & inverse functions – Graphing inverse functions – Composition of a function and its inverse – Finding the inverse of a function 2

3. One-to-one & Inverse Functions

4. 4 Recall that for each point in a function f: – Each x-coordinate is associated with only one y- coordinate Now suppose each y-coordinate is associated with only one x-coordinate: f is then called a one-to-one function f -1 is called the inverse of f and is obtained when we swap the x and y-coordinates of f e.g. f = {(0, 0), (1, 5), (-3, 4)}  f -1 = {(0, 0), (5, 1), (4, -3)}  f -1 is the inverse of f AND f is the inverse of f -1 f -1 is only NOTATION for an inverse: f -1 ≠ 1 ⁄ f f has an inverse ONLY if it is one-to-one

5. 5 One-to-one & Inverse Functions (Continued) The domain and range of a function and its inverse are switched: – e.g. f = {(0, 0), (1, 5), (-3, 4)}  f -1 = {(0, 0), (5, 1), (4, -3)}  f(domain) = {-3, 0, 1} & f(range) = {0, 4, 5}  f -1 (domain) = {0, 4, 5} & f -1 (range) = {-3, 0, 1} To test a graph to see if it is one-to-one and thus has an inverse, we use the horizontal line test – If the horizontal line crosses the graph more than once, the graph is not one-to-one A graph which initially fails the horizontal line test can have its domain restricted so that it becomes one-to-one

6. One-to-one & Inverse Functions (Example) Ex 1: Given the function f, determine i) whether it has an inverse ii) if so, state the inverse iii) if so, state the domain and range for f and f -1 a)f = {(3, 3), (-1, -5), (6, 10), (-5, -1) b)f = {(0, -4), (1, 2), (3, 5), (8, 2)} 6

7. One-to-one & Inverse Functions (Example) Ex 2: For each graph i) state whether it is one- to-one ii) if not one-to-one, what domain restrictions will make the graph one-to-one? a) b) 7

8. Graphing Inverse Functions

9. 9 Given the equation of f AND that f is a one-to-one function, we can sketch a graph of f -1 – Obtain some (x, y) coordinates using f – Swap the (x, y) coordinates of f The graph of f and its inverse f -1 is symmetric to the line y = x – If we folded a piece of paper over the line y = x, f and f -1 would lie on top of each other

10. Graphing Inverse Functions (Example) Ex 3: Graph the inverse of the function shown: 10

11. Composition of a Function and Its Inverse

12. 12 Composition of a Function and Its Inverse Given a one-to-one function f, f -1 is the inverse of f if and only if: (f ◦ f -1 )(x) = x AND The composition of f and f -1 yields the original value (f -1 ◦ f)(x) = x The composition of f -1 and f yields the original value – i.e. One undoes the effect of the other e.g. Multiplication and division are inverse functions

13. Composition of a Function and Its Inverse (Example) Ex 4: Show algebraically whether or not f(x) and g(x) are inverses: a) b) 13

14. Finding the Inverse of a Function

15. 15 Finding the Inverse of a Function Rather than a graph of f -1, we more often than not want the equation of f -1 – Recall that to get f -1 when f consisted of a set of points, we switched the x and y coordinates Given that f is a one-to-one function, to find f -1 : – Replace f(x) with y – Swap x and y – Solve for y – Substitute f -1 (x) for y Check whether the two functions are inverses Recall that the domain and range of f are reversed for f -1

16. Finding the Inverse of a Function (Example) Ex 5: i) Find f -1 ii) State the domain of f and f -1 : a) b) c) 16

17. Summary After studying these slides, you should be able to: – Identify an inverse function from a list of coordinate pairs – State the domain & range of a function and its inverse – Given points on its function, graph an inverse – Determine algebraically whether or not two functions are inverses – Find the inverse of a function Additional Practice – See the list of suggested problems for 1.9 Next lesson – Variation (Section 1.10) 17