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Inverse Functions

Definition of Inverse Functions

A function g is the inverse of the function f if

f(g(x)) x for each x in the domain of g and

g(f(x)) x for each x in the domain of f. The

function g is denoted by f-1 (read f inverse).

Graphically speaking the yellow and red

graphs are inverses of each other. See how they

mirror each other across y x.

y1/(x-2)

y x

Theorem Continuity and Differentiability of

Inverse Functions

Let f be a function whose domain is an interval

I. If f has an inverse, them the following

statements are true

1. If f is continuous on its domain, then f-1

is continuous on its domain.

2. If f is increasing on its domain, then f-1 is

increasing on its domain.

3. If f is decreasing on its domain, then f-1

is decreasing on its domain.

4. If f is differentiable at c and f-1 (c) does

not equal 0 then f-1 is differentiable at

f(c).

Theorem The Derivative of an Inverse Function

Let f be a function that is differentiable on an

interval I. If f has an inverse function g, then

g is differentiable at any x for which f (g(x))

is not zero. Moreover, g (x) 1/ f (g(x)).

Proof Since g is the inverse of f, f (g(x)) x

Taking the derivative of both sides with respect

to x, we get f (g(x))g (x) 1 Thus g (x)

1/ f (g(x))

Example

What is the value of f-1 (x) when x 3?

Since we want the inverse, 3 would be the y

coordinate of some value of x in f(x).

As you can see, we could try to guess an answer

but we have no means to solve the equation.

Lets look at the graph.

On the graph you can see that a y value of 3

corresponds to an x value of 2, thus if (2,3) is

on the f function, (3,2) is on the

function.

f-1

So, f-1 (3) 2

(2,3)

B. What is the value of (f-1) (x) when x 3?

Solution Since g (x) 1/ f (g(x)) by our

previous theorems, we can substitute f-1 for g,

thus f-1 (x) 1/ f (f-1 (x)) f-1 (3)

1/ f (f-1 (3)) 1/ f(2) 1/(3/4(2)(2)1) 1/4

Graphs of Inverse Functions Have Reciprocal Slopes

Two inverse functions are

Pick a point that satisfies f, such as (3,9),

then (9,3) satisfies g.

Homework Examples

4. Show that f and g are inverse functions (a)

algebraically and (b) graphically

Solution One way to do (a) is to show that

f(g(x))x and g(f(x)) x. A second method would

be to find the inverse of f and show that it is g.

Four steps to finding an Inverse Step 1 change

f(x) to y Step 2 Interchange x and y Step 3

solve for y Step 4 change y to f-1

Graphically f and its inverse should look like

mirror images across the line y x.

Show that f is strictly monotonic on the

indicated interval and therefore has an inverse

on that interval. (Strictly monotonic means that

f is always increasing on a given interval or f

is always decreasing on a given interval ).

The derivative is always negative on ,

therefore, f is decreasing and thus has an

inverse on this interval.

On

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