Inverse functions

Calculus5.3

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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.

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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

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Examples

• Do the following functions have inverses?

2

1

1g t

t

2

4f x

x

3 2 12f x x x 3 3f x x x

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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

1

1

f f x x

f f x x

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Example

• Find and verify the inverse of

2 3f x x

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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).

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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

1

g xf g x

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Examples

• Find (f –1)’(a).

5 32 1 2f x x x a

4 2f x x a