Math 150 – Fall 2015
Section 4F
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Section 4F – Inverse Functions
Inverse Relations
A function f pairs an output y with each input x. The inverse of f , called f −1 reverses
the process. That is, the inverse pairs the original y with the original x. That is
(a, b) ∈ f , then (b, a) ∈ f −1 .
Domain f Range f
Domain f −1 Range f −1
2
5
7
2
4
3
9
9
1
6
Theorem. The graph of f −1 is a reflection of the graph f (x) about the line y = x.
Example 1. Graph f (x) and use symmetry to graph the inverse relation. Is the inverse
relation a function f −1 (x)?
√
(a) f (x) = x − 1
(b) f (x) = x2 + 2
Math 150 – Fall 2015
Section 4F
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One-to-One Functions
Suppose both f (x1 ) = y1 and f (x2 ) = y1 . Then f −1 reverses the inputs and outputs,
so f −1 (y1 ) = x1 and f −1 (y1 ) = x2 . For f −1 to be a function, then x1 must equal x2 ,
or f −1 is not a function because the input y1 has two outputs, x1 and x2 .
This means that for a function f to have an inverse function, each x-value in the domain
must be paired with a different y-value in the range, since this y will be the input to
f −1 with output x. This is called a one-to-one function.
Definition. A function is said to be one-to-one if and only if f (x1 ) = f (x2 ) means
x1 = x2 . That is two inputs, x1 and x2 , cannot have the same output unless they were
equal to begin with.
√
Example 2. Prove algebraically the function f (x) = 7 − 3x is one-to-one.
Example 3. Determine algebraically whether the following functions are one-to-one.
If the function is not one-to-one restrict its domain so that it is one-to-one.
(a) f (x) = x2 − 3
(b) f (x) = (x − 2)3 + 4
Horizontal Line Test: A function is one-to-one if and only if no horizontal line
intersects the graph in more than one points.
Example 4. Determine if the following represents a function, a one-to-one function,
or not a function.
Math 150 – Fall 2015
Section 4F
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Definition of Inverse Function
Definition. If f is a one-to-one function with domain A and range B, then, for every
b in B, its inverse function, f −1 , is defined by f −1 (b) = a if and only if f (a) = b.
That is (a, b) ∈ f , then (b, a) ∈ f −1 .
Note.
• The domain of f −1 is the range of f .
• The range of f −1 is the domain of f .
• f ◦ f −1 (x) = x and f −1 ◦ f (x) = x
Example 5. If f is a one-to-one function, answer the following:
(a) If f (3) = 11, then f −1 (11) =
(b) If f −1 (2) = 6, then f −1 (6)=
(c) If the domain of f is [0, ∞), then the range of f −1 is
(d) If the range of f is [−5, 15], then the domain of f −1 is
√
3
Example 6. Show that f (x) = 3 3x + 5 − 4 and g(x) = x 3−5 are inverses.
Example 7. Show that f (x) =
3
x
+ 4 and g(x) =
3
x−4
are inverses
Math 150 – Fall 2015
Section 4F
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Finding the Inverse of a Function
To find the inverse of a function, replace f (x) with y. Then interchange the x’s and
y’s in the expression and then solve for y. After solving for y, replace y with f −1 (x).
Remember, for the inverse to exist, f must be a one-to-one function.
Example 8. Algebraically calculate and simplify the inverse function of the following.
Give the domain and range of f and its inverse.
√
(a) f (x) = x − 3
(b) f (x) = (x − 3)3 + 5
(c) f (x) = −2(4x − 5)2 − 7
(d) f (x) = −2(4x − 5)2 − 7, where x ∈
5
4, ∞