Chapter 5 Inverse Functions and Applications Section 5.1

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Chapter 5 Inverse Functions and
Applications Section 5.1

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Section 5.1Finding
the Inverse Numerical, Graphical, and Symbolical
Approaches

• Definition of an Inverse Function
• Determining if an Inverse is a Function
• One-to-One Functions
• Horizontal Line Test
• Finding the Inverse of a Function
• Graphs of Inverse Functions
• Applications

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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
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How can we guarantee that the inverse of the
original function will also be a function?
The original function must be a one-to-one
function! A function f is one-to-one 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 one-to-one. Let us check
several input-output values.

x 3 2 1 0 1 2 3
f(x) x3 27 8 1 0 1 8 27
Different x-values will produce different
y-values. For this function, in general, if x1 ?
x2, then (x1)3 ? (x2)3 and we can say that this
function is one-to-one.
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Determine whether the function f(x) x2 5 is
one-to-one. 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 one-to-one.

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Graphical Test Horizontal
Line Test A function is one-to-one if no
horizontal line intersects its graph more than
once. Revisiting our previous examples f(x)
x3 One-to-one function f(x) x2 5
Not one-to-one!

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Inverse
Functions If f is a one-to-one function with
ordered pairs (x, y), the inverse of f, denoted
f-1, is also a one-to-one 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 one-to-one function. The domain of f-1
is the same as the range of f, and the range of
f-1 is the same as the domain of f. Note f-1
is read as "f inverse." Caution f-1(x) is the
notation for the inverse function and it does not
mean the reciprocal of f(x).

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Points (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
one-to-one, 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).
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• Finding the Inverse of a
  Function
• Let f be a one-to-one 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 f-1(x). That is, assign the name f-1(x)
  to the resulting inverse function. 
•  
•  

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• 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 one-to-one,
  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)

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(Contd.) (4) Let y f-1(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).
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• Find the inverse function of
• Replace f(x) with y, then interchange x and y.
• Solve the new equation for y.
• 
  
• 
  
• 
  
  
• 
  

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Find the inverse of the function
This is a parabola with vertex at (0, 3). As
its graph shows next, it is not a one-to-one
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 one-to-one function.






(continued on the next slide)



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(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)



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The 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







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Given 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)



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(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.
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• 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.
• 
  

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