Title: Chapter 5 Inverse Functions and Applications Section 5.1
1 Chapter 5 Inverse Functions and
Applications Section 5.1
2 Section 5.1 Finding
the Inverse Numerical, Graphical, and Symbolical
Approaches
 Definition of an Inverse Function
 Determining if an Inverse is a Function
 OnetoOne Functions
 Horizontal Line Test

 Finding the Inverse of a Function
 Graphs of Inverse Functions
 Applications
3 Introduction to Inverse Functions Consider the
functions represented by these tables
(a) (b)
In tables (c) and (d) that
follow, we have swapped the inputs and the
outputs from (a) and (b), respectively.
(c)
(d) When we interchange the
input and the output coordinates in a function,
we are looking for the inverse of the function.
This may result in a relation that is also a
function, like examples (a) and (c). However,
the new relation may not turn out to be a
function, as in the case of (b) and (d) input
9 in (d) has 2 different outputs!
Input 3 5 8 10
Output 4 9 7 9
Input 2 3 8 13
Output 6 8 2 7
Input 6 8 2 7
Output 2 3 8 13
Input 4 9 7 9
Output 3 5 8 10
4How can we guarantee that the inverse of the
original function will also be a function?
The original function must be a onetoone
function! A function f is onetoone if for any
elements x1 and x2 in its domain, when x1 ? x2
then f(x1) ? f(x2). That is, any two different
inputs will always produce two different
outputs. Example Determine whether the
function f(x) x3 is onetoone. Let us check
several inputoutput values.
x 3 2 1 0 1 2 3
f(x) x3 27 8 1 0 1 8 27
Different xvalues will produce different
yvalues. For this function, in general, if x1 ?
x2, then (x1)3 ? (x2)3 and we can say that this
function is onetoone.
5Determine whether the function f(x) x2 5 is
onetoone. Let x1 2 and x2 2. f(2)
(2)2 5 4 5 9 f(2) (2)2 5 4
5 9 Observe that two different inputs will
produce the same output. Therefore, the given
function is not onetoone.
6 Graphical Test Horizontal
Line Test A function is onetoone if no
horizontal line intersects its graph more than
once. Revisiting our previous examples f(x)
x3 Onetoone function f(x) x2 5
Not onetoone!
7 Inverse
Functions If f is a onetoone function with
ordered pairs (x, y), the inverse of f, denoted
f1, is also a onetoone function with ordered
pairs (y, x). That is, the inverse of a
function is the set of ordered pairs obtained
when we swap the inputs and the outputs in the
original onetoone function. The domain of f1
is the same as the range of f, and the range of
f1 is the same as the domain of f. Note f1
is read as "f inverse." Caution f1(x) is the
notation for the inverse function and it does not
mean the reciprocal of f(x).
8Points (2, 0) and (1, 9) satisfy the function
f(x) 3x 6. Using this information, show that
g(x) is the inverse function of f(x)
f(x)
3x 6 is a linear function, and we know it is
onetoone, thus, its inverse will also be a
function. We know that the inverse of a
function is the set of ordered pairs obtained
when we swap the inputs and the outputs in the
original function. Therefore, we only need to
interchange the coordinates of the given points
and check that (0, 2) and (9, 1) satisfy g(x).
So, g(x) is the inverse function of
f(x).
9 Finding the Inverse of a
Function  Let f be a onetoone function defined by y
f(x).  Replace f(x) with y.
 (2) Swap the input and the output (that is,
interchange x  and y).
 (3) Solve the new equation for y. (If the
equation cannot be  solved for y, then the original function has
no inverse  function.)
 Let y f1(x). That is, assign the name f1(x)
to the resulting inverse function. 

10 Find the inverse function of f(x) 2x 8. State
the domain and range for f(x) and its inverse.  f(x) is a linear function and it is onetoone,
thus, its inverse will also be a function.  (1) Replace f(x) with y.
 y 2x 8
 Interchange x and y.
 x 2y 8
 (3) Solve the new equation for y.



(continued on the next slide)
11(Contd.) (4) Let y f1(x). The domain and
range for f(x) is (8, 8), and the domain and
range of f 1(x) is (8, 8). Optional We can
check that if a point (x, y) satisfies f(x), the
swapped coordinates will satisfy the inverse
function. Example (0, 8) satisfies f(x)
and (8, 0) satisfies f 1(x).
12 Find the inverse function of
 Replace f(x) with y, then interchange x and y.



 Solve the new equation for y.





13Find the inverse of the function
This is a parabola with vertex at (0, 3). As
its graph shows next, it is not a onetoone
function (it does not pass the horizontal line
test). In this case, we can
restrict the domain of f(x) to, lets say 0, 8),
which will guarantee a onetoone function.
(continued on the next slide)
14(Contd.) Now we can find the inverse on the
limited domain Observe the
domain of f(x) is 0, 8) and its range is 3,
8), while the domain and range of the inverse are
3, 8) and 0, 8), respectively.
(continued on
the next slide)
15The Graphs of Inverse Functions Recall that if
the graph of the original function contains a
point (a, b), then the graph of the inverse
function will contain the point (b, a). The
graph of a point (b, a) is the reflection of the
point (a, b) across the line y x. Thus, we can
summarize the following The graphs of a
function and its inverse are symmetric about the
line y x. Example
16Given the graph of f(x), graph its inverse along
with y x label the inverse g(x).
We know that the graphs of a
function and its inverse are symmetric about the
line y x. To graph the inverse we only need to
reverse the ordered pairs of the original
function. The points (2, 4), (1, 3), (0, 4),
and (1, 5) lie on the graph of f(x), therefore,
the points (4, 2), (3, 1), (4, 0), and (5, 1)
will lie on the graph of the inverse.
(continued on the next
slide)
17(Contd.) We plot the points and connect them
with a smooth curve to construct the graph of the
inverse, g(x).
Observe that both graphs are symmetric about
the line y x.
18 As of August 1, 2014, 1.00 was equivalent to
approximately  0.7447 Euros.

 a. Write a function f that represents the number
of Euros in terms of the number of dollars,
x.  b. Find the inverse of your function.



19(Contd.) c. Find and interpret
Round your answer to 2 decimal places.
In August 1, 2014, twenty Euros were
equivalent to 26.86.
20Using your textbook, practice the problems
assigned by your instructor to review the
concepts from Section 5.1.