Section 4.5 Inverse Functions NOTES
Precalculus
In the equation y = f(x), x is called the independent variable (because we can assign
it any acceptable value) and y is called the dependent variable because the value
that y obtains will depend on what the value of x is.
In some situations it might be better to switch the roles of dependent and
independent variables. For example, in the US temperatures are often reported in
0
F. Say that we have 5 temperatures: 320F, 00 F, -150F, and -400F and we want to
convert these temperatures to 0C.
For this case, what is best to use as the independent variable, 0F or 0C?
Now, say that we have 5 temperatures in 0C: 1000C, 500C, 00C, -200C, and -400C
and we want to convert these temperatures to 0F.
For this case, what is best to use as the independent variable, 0F or 0C?
One function is said to be an inverse of another function if the independent and
dependent variables are switched. One way of expressing the inverse of a function,
f, is by the notation f -1. Therefore, if g(x) is the inverse of f(x), we could express
g(x) as g(x) = f -1(x).
In relating F and if C is the independent variable, the conversion equation is
expressed as:
9
F(C) = C  32
5
For the inverse of this relation, F is the independent variable and C is the
dependent variable:
5
C F   F  32 
9
For inverse functions, if the output of one function is used as the input for its
inverse, the output for the inverse will be equal to the original input into the first
function .
For example, if the input in F(C) is 0 for C, the output is F = 32. Now if we put F
= 32 into its inverse (C(F)), the output is C = 0, which is the original input into the
first function. In terms of composition of functions, this could be expressed as
C(F(0)) = 0.
In general, if f(x) and g(x) are inverses of one another, then
f(g(x)) = x for all x in the domain of f and
g(f(x)) = x for all x in the domain of g.
This is a property that must exist for all functions that are inverses of one another,
so it is the test performed to check if two functions are inverses of one another.
For example, verify that the functions f x  
of one another.
x7
and g(x) = 14x – 7 are inverses
14
When a function f has an inverse, the graph of its inverse (f -1) can be obtained by
switching the x and y values.
For example, graph y = 3x + 1 and then, on the same plot, graph its inverse.
y
6
4
2
-6
-4
2
-2
4
6 x
-2
-4
-6
To obtain the equation for the inverse of a function, we interchange x and y and
solve the equation for y. For example, obtain the inverse of the function y = 3x + 1.
Another example: Let f(x) = 4 – x2 for x ≥ 0.
a.) Sketch the graphs of y = f(x) and y = f-1(x).
y
6
4
2
-6
-4
2
-2
4
6 x
-2
-4
-6
b.) Find a rule for f -1(x).
A function y = f(x) that has an inverse is called a one-to-one function because each
x must correspond to only 1 y value, and each y value must correspond to only 1 x
value (because in the inverse, x and y are switched). On a graph, the horizontal
line test is used to determine if a function has an inverse. To do this test, any
horizontal line drawn must intersect the graph of the function in no more than 1
point.
Example: The graphs of 2 functions are shown below. Circle the function that
will have an inverse.
Section 4.5 Inverse Functions
Precalculus
1.) Suppose a function f has an inverse. If f(2) = 6 and f(3) = 7 find:
a.) f 1  6  
b.) f 1  f  3  
c.) f  f 1  7   
2.) Suppose a function f has an inverse. If f(0) = -1 and f(-1) = 2 find:
a.) f 1  1 
b.) f 1  f  0   
c.) f  f 1  2   
3.) The graph of f  x   x3  x 2 is shown below. Explain why it has no inverse.
0.6
0.4
0.2
1
0.5
0.5
0.2
0.4
0.6
4.) If g(3) = 5 and g(-1) = 5, explain why g has no inverse.
1
x  4.
2
a.) On the graph below, sketch L(x) and L-1(x).
5.) Let L  x  
y
10
8
6
4
b.) Find an equation for L-1(x).
2
-10
-8
-6
-4
2
-2
4
6
-2
-4
-6
-8
-10
For 6 – 9 below the graph of a function is given. State whether the function has an
inverse.
6.) Does this have an inverse?
0.4
0.2
0.5
0.5
0.2
0.4
7.) Does this have an inverse?
8
10 x
y
6
8.) Does this have an inverse?
4
2
-6
-4
-2
2
4
6 x
2
4
6 x
-2
-4
-6
y
6
9.) Does this have an inverse?
4
2
-6
-4
-2
-2
-4
-6
For problems 10 – 14 state whether the function f has an inverse. If f 1 exists, find
it formula and show that f  f 1  x    f 1  f  x    x .
10.) f(x) = 3x – 5
a.) Does f 1 exist?
b.) If f 1 does exist:
What is its formula?
Show that f  f 1  x    x
Show that f 1  f  x    x
4
x
a.) Does f 1 exist?
11.) f(x) =
b.) If f 1 does exist:
What is its formula?
Show that f  f 1  x    x
Show that f 1  f  x    x
1
x2
a.) Does f 1 exist?
12.) f(x) =
b.) If f 1 does exist:
What is its formula?
Show that f  f 1  x    x
Show that f 1  f  x    x
13.) f(x) =
4  x2
a.) Does f 1 exist?
b.) If f 1 does exist:
What is its formula?
Show that f  f 1  x    x
Show that f 1  f  x    x
14.) f(x) = 3 1  x 3
a.) Does f 1 exist?
b.) If f 1 does exist:
What is its formula?
Show that f  f 1  x    x
Show that f 1  f  x    x
For 15 – 16 sketch the graphs of g and g-1. Then find a formula for g-1(x),
including the values of x (the domain) for which the formula will apply.
15.) g  x   x 2  2, x  0
y
10
8
6
4
2
2
4
6
8
x
10
-2
Formula for g-1(x) (including its domain):
y
10
16.) g  x    x  1  1, x  1
2
8
6
4
2
-10
-8
-6
-4
2
-2
-2
-4
-6
Formula for g-1(x) (including its domain):
4
6
8
10 x