6.2 Inverse Functions

1. 6.2Inverse Functions

2. A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

3. x1 y1 x1 y1 x2 y2 x2 x3 x3 y3 y3 Domain Range Domain Range One-to-one function NOT One-to-one function x1 y1 y2 x3 y3 Not a function Domain Range

4. Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

5. Use the graph to determine whether the function is one-to-one. Not one-to-one.

6. Use the graph to determine whether the function is one-to-one. One-to-one.

7. Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f-1(f( x )) = x for every x in the domain of f and f(f-1(x))=x for every x in the domain of f-1. .

8. Apply f Apply f-1 Input x f(x) f-1(f( x ))=x Apply f-1 Apply f Input x f(f-1(x))=x f-1(x)

9. Domain of f Range of f

10. Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

11. y = x (0, 2) (2, 0)

12. Procedure for Finding the Inverse of a One-to-One Function • In y = f(x) interchange the variables x and y to obtain x = f(y) • If possible, solve the implicit equation for y in terms of x to obtain the explicit form of f-1. • Check the results by showing that f-1(f( x )) = x and f(f-1(x)) =x y= f-1(x)

13. Find the inverse of The function is one-to-one. Interchange variables. Solve for y.

14. Check.