Section 4.2: Inverse Functions
Definition: A function f with domain A is called one-to-one if no two elements of A have
the same image. That is,
f (x1 ) = f (x2 ) if and only if x1 = x2 .
For instance, f (x) = x2 is not one-to-one since f (−1) = 1 = f (1) and −1 6= 1.
Theorem: (Horizontal Line Test)
A function f is one-to-one if and only if no horizontal line intersects the graph of y = f (x)
more than once.
Example: Prove that f (x) = x2 − 2x + 5 is not one-to-one.
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Definition: Let f be a one-to-one function with domain A and range B. Then its inverse
function f −1 has domain B and range A and is defined by
f (x) = y ⇐⇒ f −1 (y) = x.
Moreover, the inverse function satisfies
f −1 (f (x)) = f (f −1 (x)) = x.
How to Find the Inverse of a One-to-One Function:
Suppose that f is a one-to-one function.
1. Let y = f (x).
2. Interchange x and y.
3. Solve this equation for y. The solution is y = f −1 (x).
Example: For each function, show that f is one-to-one and find its inverse.
(a) f (x) = 5 − 4x3
2
(b) g(x) =
(c) f (x) =
√
2 + 5x
x−3
x+3
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Note: The graph of f −1 is obtained by reflecting the graph of f about the line y = x.
Example: The graph of a function f is given below. Sketch the graph of f −1 .
Theorem: (Derivative of an Inverse Function)
Suppose that f is a one-to-one differentiable function with inverse f −1 . Then the inverse
function is differentiable and
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(f −1 )0 (x) = 0 −1
.
f (f (x))
Example: If f (4) = 5 and f 0 (4) = 2, find (f −1 )0 (5).
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Example: Find (f −1 )0 (a) for each function.
(a) f (x) = x5 − x3 + 2x, a = 2
(b) f (x) =
√
x3 + x2 + x + 1, a = 2
(c) f (x) = 3 + x + ex , a = 4
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