Download 171S5.1 - Cape Fear Community College

MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 5:
Exponential and
Logarithmic Functions
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
5.1 Inverse Functions
· Determine whether a function is one-to-one, and if it is,
find a formula for its inverse.
· Simplify expressions of the type
and
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.
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
Interchanging the first and second coordinates of each ordered pair in
a relation produces the inverse relation.
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.
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 in g across the line
y = x.
Inverse Relation
If a relation is defined by an equation, interchanging the
variables produces an equation of the inverse relation.
This graphing program will show the graph f(x) and trace its reflection about a straight
line. http://cfcc.edu/mathlab/geogebra/function_reflection_line.html
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.
One-to-One Functions
A function f is one-to-one if different inputs have
different outputs – that is,
if
a ≠ b,
then
f (a) ≠ f (b).
Or a function f is one-to-one if when the outputs are the
same, the inputs are the same – that is,
if
f (a) = f (b),
then
a = b.
Inverses of Functions
If the inverse of a function f is also a function, it is named f -1 and
read “f-inverse.”
The –1 in f -1 is not an exponent.
f -1 does not mean the reciprocal of f and f -1(x) can
not be equal to
One-to-One
Functions and Inverses
·If a function f is one-to-one, then its inverse f -1 is a function.
·The domain of a one-to-one function f is the range of the inverse f -1.
·The range of a one-to-one function f is the domain of the inverse f -1.
·A function that is increasing over its domain or is decreasing over its domain
is a one-to-one function.
Horizontal-Line Test
If it is possible for a horizontal line to intersect the graph of a
function more than once, then the function is not one-to-one and its
inverse is not a function.
not a one-to-one function
inverse is not a function
Example
From the graph shown, determine whether each function is one-to-one
and thus has an inverse that is a function.
No horizontal line intersects more
than once: is one-to-one; inverse is
a function.
Horizontal lines intersect more than
once: not one-to-one; inverse is not
a function.
Example
From the graph shown, determine whether each function is one-to-one
and thus has an inverse that is a function.
No horizontal line intersects more
than once: is one-to-one; inverse is
a function
Horizontal lines intersect more than
once: not one-to-one; inverse is not
a function
IMPORTANT
Obtaining a Formula for an Inverse
If a function f is one-to-one, 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).
Example
Determine whether the function f (x) = 2x - 3 is one-to-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 one-to-one
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
4. Replace y with f -1(x):
Example
Graph
using the same set of axes. Then compare the two
graphs.
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.
Example (continued)
The graph f -1 is a reflection of the graph f across the line y =
x.
Inverse Functions and Composition
If a function f is one-to-one, then f -1 is the unique function such
that each of the following holds:
for each x in the
domain of f, and
for each x in the
domain of f -1.
Example
Given that f (x) = 5x + 8, use composition of functions to show that
Solution:
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.
This is not the equation of a function
because an input of 4 would yield two
outputs, - 2 and 2.
Restricting a Domain
However, if we restrict the domain of f (x) = x2 to nonnegative
numbers, then its inverse is a function.
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.
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
389/20. Given the function f, prove that f is one-to-one using the definition of a
one-to-one function on p. 382.
f(x) = ∛x
389/24. Given the function g, prove that g is not one-to-one using the definition of
a one-to-one function on p. 382.
g(x) = 1 / x6
390/__. Using the horizontal-line test, determine whether the function is one-toone.
390/40. Graph the function and determine whether the function is one-to-one
using the horizontal-line test.
390/42. Graph the function and determine whether the function is one-to-one
using the horizontal-line 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 one-to-one.
b) If the function is one-to-one, find a formula for the inverse.
391/80. The graph represents a one-to-one function f. Sketch the graph of the
inverse function f -1 on the same set of axes.
391/88. For the function f , use composition of functions to show that f -1 is as
given.
391/94. Find the inverse of the given one-to-one 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
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