Page 1
Math 151-copyright Joe Kahlig, 14A
Section 4.2: Inverse Functions
Definition: A function of domain A is said to be a one-to-one function if no two elements of A
have the same image; that is
If x1 6= x2 then f (x1 ) 6= f (x2 )
An alternate statement of one-to-one is the contrapositive of the above statement:
If f (x1 ) = f (x2 ) then x1 = x2
Example: Show that these functions are one-to-one.
A) y =
√
B) y =
x−2
x+2
x+5
Definition: Let f be a one-to-one function with domain A and range B. Then the inverse function
f −1 has the domain B and range A and is defined for any y in B by
f −1 (y) = x ⇔ f (x) = y
Example: If f (x) is a one-to-one function and f (x) = y then
f −1 (f (x)) =
f (f −1 (y)) =
Example: Find the inverse function of f . State the domain and range of f −1 .
f (x) =
x−2
x+2
Page 2
Math 151-copyright Joe Kahlig, 14A
Example: Find the inverse function of f (x) =
f (x) =
√
√
x − 5. State the domain and range of f −1 .
x−5
Theorem: If f is a one-to-one differentiable function with inverse function g = f −1 and f ′ (g(a)) 6= 0,
then the inverse function is differentiable at a and
g′ (a) =
1
f ′ (g(a))
Example: If g is the inverse function to f , compute g′ (0) and g ′ (−8)
x−4
f (x) =
x+5
Example: If g is the inverse function to f , compute g′ (6).
f (x) = 5 + 2x + e3x