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171S5.1 Inverse Functions
November 04, 2010
5.1 Inverse Functions
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 5: Exponential and Logarithmic Functions
• Determine whether a function is one­to­one, and if it is, find a formula for its inverse.
• Simplify expressions of the type
and
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest
Inverses
When we go from an output of a function back to its input or inputs, we get an inverse relation. When that relation is a function, we have an inverse function.
Example
Consider the relation g given by g = {(2, 4), (–1, 3), (­2, 0)}.
Graph the relation in blue. Find the inverse and graph it in red.
Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation.
Consider the relation h given as follows: h = {(­8, 5), (4, ­2), (­7, 1), (3.8, 6.2)}.
The inverse of the relation h is given as follows:
{(5, ­8), (­2, 4), (1, ­7), (6.2, 3.8)}.
Inverse Relation
Solution: The relation g is shown in blue. The inverse of the relation is
{(4, 2), (3, –1), (0, ­2)} and is shown in red. The pairs in the inverse are reflections of the pairs Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation.
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171S5.1 Inverse Functions
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Inverse Relation
If a relation is defined by an equation, interchanging the variables produces an equation of the inverse relation.
Example
Find an equation for the inverse of the relation:
y = x2 ­ 2x.
Solution: We interchange x and y and obtain an equation of the inverse:
x = y2 ­ 2y.
Graphs of a relation and its inverse are always reflections of each other across the line y = x.
Graphs of a Relation and Its Inverse
If a relation is given by an equation, then the solutions of the inverse can be found from those of the original equation by interchanging the first and second coordinates of each ordered pair. Thus the graphs of a
relation and its inverse are always reflections of each other across the line y = x.
One­to­One Functions
A function f is one­to­one if different inputs have different outputs – that is,
if a ≠ b, then f (a) ≠ f (b).
Or a function f is one­to­one if when the outputs are the same, the inputs are the same – that is,
if f (a) = f (b), then a = b.
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Inverses of Functions
One­to­One Functions and Inverses
If the inverse of a function f is also a function, it is named f ­1 and read “f­inverse.”
The –1 in f ­1 is not an exponent.
• If a function f is one­to­one, then its inverse f −1 is a function.
• The domain of a one­to­one function f is the range of the inverse f −1.
f ­1 does not mean the reciprocal of f and f ­1(x) can
• The range of a one­to­one function f is the domain of the inverse f −1.
not be equal to
• A function that is increasing over its domain or is decreasing over its domain is a one­to­one function.
Horizontal­Line Test
Example
If it is possible for a horizontal line to intersect the graph of a function more than once, then the function is not one­to­one and its inverse is not a function.
not a one­to­one function
inverse is not a function
From the graph shown, determine whether each function is one­to­one and thus has an inverse that is a function.
No horizontal line intersects more than once: is one­to­one; inverse is a function.
Horizontal lines intersect more than once: not one­to­
one; inverse is not a function.
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171S5.1 Inverse Functions
Example
From the graph shown, determine whether each function is one­to­one and thus has an inverse that is a function.
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Obtaining a Formula for an Inverse
If a function f is one­to­one, a formula for its inverse can generally be found as follows:
1. Replace f (x) with y.
2. Interchange x and y.
3. Solve for y.
4. Replace y with f −1(x).
No horizontal line intersects more than once: is one­to­one; inverse is a function
Horizontal lines intersect more than once: not one­to­
one; inverse is not a function
Example
Example
Determine whether the function f (x) = 2x − 3 is one­to­
one, and if it is, find a formula for f −1(x).
Solution: The graph is that of a line and passes the horizontal­line test. Thus it is one­to­one and its inverse is a function.
1. Replace f (x) with y:
y = 2x ­ 3
2. Interchange x and y:
x = 2y ­ 3
3. Solve for y: x + 2 = 3y
Graph
using the same set of axes. Then compare the two graphs.
4. Replace y with f ­1(x):
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Example (continued)
Example (continued)
Solution: The solutions of the inverse function can be found from those of the original function by interchanging the first and second coordinates of each ordered pair.
The graph f −1 is a reflection of the graph f across the line y = x.
Inverse Functions and Composition
−1
If a function f is one­to­one, then f is the unique function such that each of the following holds:
Example
Given that f (x) = 5x + 8, use composition of functions to show that
Solution:
for each x in the domain of f, and
for each x in the domain of f ­1.
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171S5.1 Inverse Functions
Restricting a Domain
When the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function. In such cases, it is convenient to consider “part” of the function by restricting the domain of f (x).
Suppose we try to find a formula for the inverse of f (x) = x2.
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Restricting a Domain
However, if we restrict the domain of f (x) = x2 to nonnegative numbers, then its inverse is a function.
This is not the equation of a function because an input of 4 would yield two outputs, ­ 2 and 2.
389/4. Find the inverse of the relation:
{(­1, 3), (2, 5), (­3, 5), (2, 0)}
389/8. Find an equation of the inverse relation of
y = 3x2 ­ 5x + 9.
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171S5.1 Inverse Functions
389/14. Graph the equation by substituting and plotting points. Then reflect the graph across the line y = x to obtain the graph of its inverse. x = ­y + 4
389/24. Given the function g, prove that g is not one­to­one using the definition of a one­to­one function on p. 382.
g(x) = 1 / x6
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389/20. Given the function f, prove that f is one­to­one using the definition of a one­to­one function on p. 382.
f(x) = ∛x
390/__. Using the horizontal­line test, determine whether the function is one­to­one.
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390/40. Graph the function and determine whether the function is one­to­one using the horizontal­line test.
390/42. Graph the function and determine whether the function is one­to­one using the horizontal­line test.
390/52. Graph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of f and of f ­1.
f(x) = 3 ­ x2, x ≥ 0
390/68. For the function: f(x) = 4x2 + 3, x ≥ 0
a) Determine whether it is one­to­one.
b) If the function is one­to­one, find a formula for the inverse.
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171S5.1 Inverse Functions
391/80. The graph represents a one­to­one function f. Sketch the graph of the inverse function f ­1 on the same set of axes.
391/94. Find the inverse of the given one­to­one function f. Give the domain and the range of f and of f ­1 , and then graph both f and of f ­1 on the same set of axes. f(x) = ∛(x) ­ 1
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391/88. For the function f , use composition of functions to show that f ­1 is as given.
392/101. Reaction Distance. Suppose you are driving a car when a deer suddenly darts across the road in front of you. During the time it takes you to step on the brake, the car travels a distance D, in feet, where D is a function of the speed r, in miles per hour, that the car is traveling when you see the deer. That reaction distance D is a linear function given by
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