# Inverse Functions

DEFINITION: If the functions f and g satisfy

the two conditions

g(f(x)) = x for every x in the domain of f

f(g(y)) = y for every y in the domain of g

(*)

then we say that f and g are inverses. Moreover,

we call **f an inverse function for g **and

g an inverse function for f.

NOTATION: The inverse of a function f is commonly

denoted by f^{-1}.

So, we can reformulate (*) as

f^{-1}(f(x)) = x for every x in the domain of f

f(f^{-1}(x)) = x for every x in the domain of f^{-1}

1. Let f(x) = x^{3}, then
since

2. Let f(x) = x^{3} + 1, then

since

and

3. Let f(x) = 2x, then since

and

4. Let f(x) = x, then f^{-1}(x) = x, since

f^{-1}(f(x)) = x

and

f(f^{-1}(x)) = x.

5. Let f(x) = 7x + 2, then

since

and

IMPORTANT:

domain of f^{-1} = range of f

range of f^{-1} = domain of f

1. Let
,
then f^{-1}(x) = x^{2}, x≥0.

2. Let then

3. Let then

4. Let then

THEOREM (The Horizontal Line Test ): A

function f has an inverse function if and only if

its graph is cut at most once by any horizontal

line .

1. The functions

are not invertible.

2. Let f(x) = x^{2}, x≥0. Then

3. Let f(x) = x^{2}, x≥2. Then

4. Let f(x) = x^{2}, x < −3. Then

5. The function f(x) = x^{2}, x > −1 is not

invertible.

THEOREM: If f has an inverse function f^{-1},

then the graphs of y = f(x) and y = f^{-1}(x) are

reflections of one another about the line y = x;

that is, each is the mirror image of the other with

respect to that line .

THEOREM: If the domain of a function f is an

interval on which f'(x) > 0 or on which f'(x) <

0, then f has an inverse function.

1. The function f(x) = x^{5}+x+1 is invertible,

since f'(x) = 5x^{4} + 1 > 1.

THEOREM: Suppose that f is a function with

domain D and range R. If D is an interval and f

is continuous and one-to-one on D, then R is an

interval and the inverse of f is continuous on R.

THEOREM(** Differentiability of Inverse Functions**):

Suppose that f is a function whose domain

D is an open interval, and let R be the

range of f. If f is differentiable and one -to-one

on D, then f^{-1} is differentiable at any value x

in R for which f'(f^{-1}(x)) ≠ 0. Furthermore, if

x is in R with f'(f^{-1}(x)) ≠ 0, then

COROLLARY: If the domain of a function f is an

interval on which f'(x) > 0 or on which f'(x) <

0, then f has an inverse function f^{-1} and f^{-1}(x)

is differentiable at any value x in the range of f.

The derivative of f^{-1} is given by formula (**).

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