Inverse functions Calculus 5.3

2 Inverse functions Switch x and y coordinates Switch domains and ranges Undo each other. Not all functions have an inverse, but if they do it is unique. Graphs of inverse functions have reciprocal slopes.

3 One-to-one functions For every y there is only one x For every x there is only one y Pass the horizontal line test Strictly monotonic (inc. or dec. over domain) Derivative is always positive or always negative Only functions that have inverses

4 Examples Do the following functions have inverses?

5 Finding an inverse algebraically Interchange x and y Solve for y Domain of inverse is range of original function Check that functions undo each other

6 Example Find and verify the inverse of

7 Theorem 5.8 If f is continuous on its domain, then f –1 is continuous on its domain. If f is increasing on its domain, then f –1 is increasing on its domain. If f is decreasing on its domain, then f –1 is decreasing on its domain. If f is differentiable at c and f’(c) ≠ 0, then f –1 is differentiable at f(c).

8 Theorem 5.9 Let f be a function that is differentiable on an interval I. If f possesses and inverse function g, then g is differentiable at any x for which f’(g(x)) ≠ 0 and

9 Examples Find (f –1 )’(a).