Page 1 | © 2011 by Janice L. Epstein 4.2 Inverse Functions Page 2 | © 2011 by Janice L. Epstein 4.2 Inverse Functions Inverse Functions (Section 4.2)
EXAMPLE 2
How can we restrict the domain of f ( x ) = cos x to make it one-toone?
A function f(x) is one-to-one provided that whenever
f ( x1 ) = f ( x2 ) then x1 = x2.
EXAMPLE 1
Which of the following functions are one-to-one?
(a) f ( x) = x 2 - 2 x + 5
(b) f ( x) = 5 - x3
Definition: Let f(x) be a one-to-one function with domain D and
range R. Then the inverse function f -1 ( x) exists. The domain of
f -1 ( x) is R and the range of f -1 ( x) is D. Moreover,
f ( x) = y  f -1 ( y ) = x
EXAMPLE 3
Find the inverse, and the domain and range of the following
functions
x-2
(c) f ( x) =
x+2
5
4
3
2
1
-4
-3
-2
-1
1
2
-1
-2
-3
-4
(a)
-5
(b) f ( x) = 5 - 4 x3
3
4
Page 3 | © 2011 by Janice L. Epstein (c) f ( x) =
4.2 Inverse Functions Page 4 | © 2011 by Janice L. Epstein 4.2 Inverse Functions 2 x +1
1- 3 x
Theorem: If f is a one-to-one differentiable function with inverse
1
g = f -1 , the g is differentiable and g ¢(a) =
f ¢ ( g (a ))
EXAMPLE 4
Suppose g is the inverse of f and
1
f (2) = 3, f ¢(2) = 7, f (3) = 4, and f ¢(3) = , find g ¢(3) .
2
(d) f ( x) = x 2 + x, x ³ -
1
2
EXAMPLE 5
Suppose g is the inverse of f. Find g ¢(4) if f ( x) = 3 + x + e x .
EXAMPLE 6
Suppose g is the inverse of f. Find g ¢(2) if
f ( x) = x3 + x 2 + x + 1 .