Calculus Section 5.3 Inverse Functions
-Verify that one function is the inverse function of another
-Determine whether a function has an inverse
-Find the derivative of an inverse function
Homework: page 347 #’s 1, 3, 9-16, 29, 32,
33, 37, 38, 71, 73
Given a set of points for the function f: {(1,4), (2,5), (3,6), (4,7)}, the inverse is f -1: { _____________________}.
So, the domain of f is the range of g, and the range of f is the domain of g.
These two functions have the effect of “un-doing” the other function. In other words, when you compose
them as f with f -1 or as f -1 with f, we have the following:
f(f -1(x)) = x
and
f -1(f(x)) = x.
Definition of the Inverse of a Function
A function g is the inverse function of the function f if:
and if:
The function g is denoted by f -1 (f-inverse).
If g is the inverse of f, then f is __________________________.
A function does not have to have an inverse function, but if it does, the inverse function is ___________.
To find an inverse, take the opposite operation from the given function. For example, if a function has
addition, you ________________. If a function has multiplication, you would _____________________.
Example)
f(x) = 2x3 – 1
What happens to x (in words): _____________________________________________________________.
Then, the inverse is the opposite: ___________________________________________________________.
f -1 = g =
Verify you answer:
f(g(x))
g(f(x))
alternate method to find f -1:
Existence of an Inverse Function
1) A function has an inverse if and only if it is one-to-one
2) If f is strictly monotonic (______________________________________) on its entire domain, then it is oneto-one and therefore has an inverse function.
__________________ Line Test
The graph of a function can help you determine whether a function is one-to-one or not. If any horizontal line
crosses a function more than once, then the function fails the horizontal line test and isn’t one-to-one (this is
similar to the vertical line test).
x3 + x – 1
x3 – x – 1
Properties on a Function and Its Inverse
1) If f is continuous on its domain, then f -1 is __________________ on its domain.
2) If f is increasing on its domain, then f -1 is __________________ on its domain.
3) If f is decreasing on its domain, then f -1 is __________________ on its domain.
4) If f is differentiable on an interval containing c and f '(c)  0 , then f -1 is __________________ at f(c).
5) The graph of f contains the point (a,b) if and only if the graph of f -1 contains the point __________.
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
1
at any x for which f '( g ( x))  0 , and g '( x) 
.
f '( g ( x))
Example
Let f ( x) 
1 3
x  x  1 . a) What is the value of f -1(x) when x = 3? b) What is the value of (f -1)’(x) when x = 3?
4