Inverse Functions

Inverse Functions (Definition)

Let f(x) be a 1-1 function then g(x) is an inverse function of f(x) if

 f(g(x)) = g(f(x)) = x

Example:

For

f(x) = 2x - 1

f -1(x) = 1/2 x +1/2

Since

f(f -1(x) ) = 2[1/2 x +1/2] - 1 = x

and

f -1(f(x)) = 1/2 [2x - 1] + 1/2 = x

The Horizontal Line Test and Roll's Theorem

Note that if f(x) is differentiable and the horizontal line test fails then

f(a)  =  f(b)

and Rolls theorem implies that there is a c such that

f '(c) = 0

A partial converse is also true:

 TheoremIf f is differentiable and f '(x) is always non negative (or always non positive) then f(x) has an inverse.

Example:

f (x) = x3 + x - 4

has an inverse since

f'(x) = 3x2 + 1

which is always positive.

Continuity and Differentiability of the Inverse Function

 Theorem f  continuous implies that f -1 is continuous. f  increasing implies that  f -1 is increasing. f decreasing implies that f -1 is decreasing. f differentiable at c and f '(c)  is not 0 implies that f -1 is differentiable at f (c). If g(x) is the inverse of  the differentiable f(x) then                               1  g'(x) =                                 f '(g(x))  if f '(g(x)) is not 0.

Proof  of (5)

Since

f (g(x)) = x

we differentiate implicitly:

d                             d
f (g(x))    =             x
dx                           dx

Using the chain rule

y = f(u),  u = g(x)

dy           dy        du
=
dx            du       dx

=   f '(u)  g'(x)   =   f '(g(x))  g'(x)

So that

f '(g(x))  g'(x)   =   1

Dividing, we get: Example:

For x > 0,  let

f(x) = x2

and be its inverse, then Note that Exercises:

1. Let

f(x) = x3 + x - 4

Find

d/dx[f -1(-4)] 2. Let Find

d/dx[f -1(0)] 