Math 127 - Section 8.
4 - Page 1
Section 8.4 - Composite and Inverse Functions
I.
Composition of Functions
A. If f and g are functions, then the composite function of f and g (written f
g) is:
(f g)(x) = f(g(x))
The domain of f
g is the set of all x in the domain of g such that g(x) is in the domain of f.
B.
With composition, we are, in effect, substituting a number into g(x), finding out what y is, and
then substituting that answer into f(x).
C.
Examples - Let f(x) = 9 2x, g(x) = 5x + 2. find the following.
1. (f g)(x)
First, we use the definition of composition to get:
(f g)(x) = f(g(x))
Now we will substitute into this equation what g(x) is equal to:
(f g)(x) = f(5x + 2)
Next, we substitute 5x + 2 in for x in f(x), EVEN THOUGH x IS REPEATED!
(f g)(x) = 9 2(5x + 2)
Simplifying, we get:
Answer: (f g)(x) = 5 + 10x
2.
Now you try one: (g f)(x)
Answer: (g f)(x) = 43 + 10x
Note that composition, in general, is not commutative.
3.
(f
g)(3)
(f
g)(3) = f(g(3))
(f
g)(3) = f(5(3) + 2) = f(13)
(f
g)(3) = 9 2(13)
Again, we start by using the definition of composition to get:
Substituting 3 for x in g(x), we get:
We now substitute 13 in for x in f(x) to get:
Simplifying, we get:
Answer: (f g)(3) = 35
4.
Now you try one: (g f)(3)
Answer: (g f)(3) = 13
Copyright 2010 by John Fetcho. All rights reserved.
Math 127 - Section 8.4 - Page 2
5.
y = f(x)
a.
(f
g)( 2)
(f
g)( 2) = f(g(2))
(f
g)( 2) = f(2)
y = g(x)
We start by using the definition of composition:
We now have to determine the value of y when x is 2 for the graph of g(x):
We next look at the graph of f(x) and determine the value of y when x is 2:
Answer: (f g)( 2) = 3
b.
II.
Now you try one: (g f)( 4)
Answer: (g f)( 4) = 2
Inverse Properties
A.
Recall that for a real number A, the additive inverse was that real number B such that
A + B = 0.
B.
For a real number A 0, the multiplicative inverse is that real number B such that AB = 1.
C.
For a function f(x), the inverse function is that function g(x) such that ( f g ) (x) = x and
D.
(g f ) (x) = x.
Verifying that functions are inverses of each other.
1. Do the composition ( f g ) (x). If the answer is x, you are halfway there.
Now do the composition (g f ) (x). If this answer is also x, then f and g are inverse
-1
functions of each other. We then would write that g(x) = f (x). The "-1" is NOT an
exponent. This notation means that we have the inverse function of f(x). Note that f(x) is
-1
also g (x).
Examples - Determine if f(x) and g(x) are inverses of each other.
3
1. f(x) = 3 x 4 , g(x) = x + 4
We first do ( f g ) (x).
2.
E.
( f g ) (x) = f(g(x)) = f(x3 + 4)
( x 3 + 4) 4 =
x3 + 4 4 =
(g f ) (x) = g(f(x)) = g(
Now substitute this in for x in f.
3
So this is half right. Now we do (g f ) (x).
x3 = x
x4 )=( x4 ) +4=x-4+4=x
3
Answer: f(x) and g(x) are inverses.
Copyright 2010 by John Fetcho. All rights reserved.
Math 127 - Section 8.4 - Page 3
2.
Now you try one:
f(x) = 5x 9, g(x) =
x+5
9
Answer: f(x) and g(x) are not inverses.
III.
Inverse Functions
A.
B.
For a function f(x), the inverse function is that function g(x) such that ( f g ) (x) = x and
(g f ) (x) = x.
Verifying that functions are inverses of each other.
1. Do the composition ( f g ) (x). If the answer is x, you are halfway there.
2.
Now do the composition (g f ) (x). If this answer is also x, then f and g are inverse
functions of each other. We then would write that g(x) = f 1 (x). The "1" is NOT an
exponent. This notation means that we have the inverse function of f(x). Note that f(x) is
also g 1 (x).
IV. Determining if a Function has an Inverse
A.
A function f(x) is one-to-one if for every y in the range there is only one x in the domain that
corresponds to it.
B.
Horizontal Line Test: A function f(x) is not one-to-one if any horizontal line intersects the
graph of f(x) in more than one point.
C.
If a function f(x) is one-to-one, then its inverse is also a function. When this occurs, we write the
-1
inverse function as f (x) (read "f inverse of x"). Note that this is the functional inverse, NOT
the multiplicative inverse.
D.
Examples - Are these functions one-to-one?
1.
Answer: Not one-to-one.
2.
Answer: Yes one-to-one.
Copyright 2010 by John Fetcho. All rights reserved.
Math 127 - Section 8.4 - Page 4
3.
Now you try one:
Answer: Yes one-to-one.
V.
Finding the Inverse
A.
In general, to find the inverse of a relation, we switch x & y in the ordered pairs. Remember that
x is the domain, y is the range.
B.
This means, geometrically, that the graph of a relation and its inverse are reflections of each
other across the identity function line, f(x) = x.
C.
Finding the inverse of a function f(x)
1. Determine if the function is one-to-one.
2. Write y for f(x).
3. Switch x & y.
4. Solve for y.
5. Write f 1 (x) for y.
6.
7.
D.
Verify by showing that f f 1 ( x) = f 1 f ( x ) = x .
Remember:
a. Domain of f is the range of f 1.
b. Range of f is the domain of f 1.
Examples - Find the inverse function.
1. f(x) = 4x 5
First, we write y for f(x).
y = 4x 5
Next, switch x & y.
x = 4y 5
Now we solve for y.
x+5
x + 5 = 4y
OR
= y
4
1
5
1
Answer: f (x) = x +
4
4
Verify:
5
5
1
1
f f 1 ( x) = f(f 1 (x)) = f x + = 4 x + 5 = x + 5 5 = x So this is ok.
4
4
4
4
You verify that f 1 f ( x) = x .
Note that if we graph these, we make a table for the function that is the "easiest", then
switch x & y to get the table for the inverse.
Copyright 2010 by John Fetcho. All rights reserved.
Math 127 - Section 8.4 - Page 5
2.
f(x) = 3 x 5
First, we replace f(x) with y.
y = 3 x5
Now we switch x & y.
x = 3 y5
Now we solve for y.
3
3
x =y5
OR
x +5=y
3
1
Answer: f (x) = x + 5
Verify:
( f f 1)( x) = f(f
You verify that f 1 f ( x) = x .
3.
Now you try one:
3
3
(x)) = f(x + 5) = 3 x3 + 5 5 = x3 = x. So this is ok.
f(x) = x + 1
1
Answer: f (x) = 3 x 1
VI. Graphing a function and its inverse.
A. Remember that to find the inverse, we switch x & y.
B. So if we make a table for f(x), to get a table for f 1 (x), we just switch x & y on the table.
C. Examples - Graph f(x) and f 1 (x) on the same set of axes.
1
5
1. f(x) = 4x 5, f 1 (x) = x +
4
4
Making a table for f(x) will be relatively easy, but f 1 (x) doesn't look so nice!
x
0
1
f(x) = 4x 5
5
1
Now graph both of these.
Switch x & y on the table
to get the table for f 1 (x).
f(x)
f (x) =
5
1
0
1
f 1 (x)
Copyright 2010 by John Fetcho. All rights reserved.
1
5
x+
4
4
