Chapter 2.7
Function Operations and Composition
Arithmetic Operations on Functions
As mentioned near the end of Section 2.3,
economists frequently use the equation “profit
equals revenue minus cost,” or
P(x) = R(x) – C(x),
where x is the number of items produced and sold.
That is, the profit function is fund by subtracting
the cot function from the revenue function.
Figure 94 shows the situation for a company that
manufactures DVDs.
The two lines are the graphs of the linear functions
for revenue R(x) = 168x
and Cost C(x) = 118x +800
with x, R(x) and C(x) given in thousands
Example 1 Stretching or Shrinking a Graph
Graph each
function
y
f x  x
x
x
-2
-1
0
1
2
|x|
When 30,000 DVDs are produced and sold, profit
is
P(30) = R(30) – C(30)
= 168(30) – [118(30) + 800]
= 5040 – 4340
= 700
That is the profit from the
sale of 30,000 DVDs is
$700
Example 1 Using Operations on Functions
Let f(x) = x2 + 1 and g(x) = 3x + 5
(a) (f + g)(1)
Example 1 Using Operations on Functions
Let f(x) = x2 + 1 and g(x) = 3x + 5
(b) (f - g)(-3)
Example 1 Using Operations on Functions
Let f(x) = x2 + 1 and g(x) = 3x + 5
(c) (f g)(5)
Example 1 Using Operations on Functions
Let f(x) = x2 + 1 and g(x) = 3x + 5
f
(d )  0
g
Example 2 Using Operations on Functions and Determining Domains
Let f(x) 8x - 9 and g(x) 2x -1
( a)
 f  g x
Example 2 Using Operations on Functions and Determining Domains
Let f(x) 8x - 9 and g(x) 2x -1
(b)
 f  g x
Example 2 Using Operations on Functions and Determining Domains
Let f(x) 8x - 9 and g(x) 2x -1
(c)
 fg x 
Example 2 Using Operations on Functions and Determining Domains
Let f(x) 8x - 9 and g(x) 2x -1
f
(d )  x 
g
Example 2 Using Operations on Functions and Determining Domains
Let f(x) 8x - 9 and g(x) 2x -1
(e) Give the domains of the functions in parts
(a) – (d).
Example 3 Evaluating Combinations of Functions
If possible, use the given representations of functions f
and g to evaluate
 f  g 4
 f  g  2
 fg 1
f
 0
g
Example 3 Evaluating Combinations of Functions
If possible, use the given representations of functions f
and g to evaluate
 f  g 4
 f  g  2
 fg 1
f
 0
g
Example 3 Evaluating Combinations of Functions
If possible, use the given representations of functions f
and g to evaluate
 f  g 4
 f  g  2
 fg 1
f
 0
g
The Difference Quotient
Suppose the point P lies on the graph of y = f(x),
and h is a positive number.
If we let (x, f(x)) denot the coordinates of P and
(x+h, f(x+h)) denote the coordinates of Q, then the
line joining P and Q has slope
f x  h   f x 
f x  h   f x 
m

,h  0
x  h   x
h
This difference is called the difference quotient.
Figure 96 shows the graph of the line PQ (called a
secant line.
As h approaches 0, the slope of this secant line
approaches the slope of the line tangent to the
curve at P. Important applications of this idea are
developed in calculus.
The next example illustrates a three-step process
for finding the difference quotient of a function.
Example 4 finding the Difference Quotient
Let f(x) = 2x2 – 3x. Find the difference quotient
and simplify the expression.
Step 1. Find f(x + h)
Step 2. Find f(x + h) – f(x)
Step 3. Find the difference quotient.
f x  h   f x 
h
Composition of Functions
The diagram in Figure 97 shows a function f that
assigns to each x in its domain a value f(x).
Then another function g assigns to each f(x) in its
domain a value g[f(x)]. This two step process takes
an element x and produces a corresponding element
g[f(x)].
T hefunction with y - values g[f(x)]
is called thecomposition of functions
g and f, writteng  f.
As a real-life example of function composition,
suppose an oil well off the California coast is
leaking, with a leak spreading iol in a circular
layer over the water’s surface.
At any time t, in minutes, after the beginning of
the leak, the radius of the circular oil slick is
r(t) = 5t feet.
Since A(r)   r gives thearea of a
2
circle of radius r, thearea can be
expressedas a functionof timein by
substituting 5r for r in A(r)   r
to get
Art    5t  25π t
2
2
2
Example 5 Evaluating Composite Functions
4
Let f(x) 2x - 1 and g(x) 
,
x 1
Find each composition  f  g 2
Example 5 Evaluating Composite Functions
4
Let f(x) 2x - 1 and g(x) 
,
x 1
Find each composition g  f  3
Example 5 Evaluating Composite Functions
4
Let f(x) 2x - 1 and g(x) 
,
x 1
Find thedomain of g  f
Example 6 Finding Composition Functions
Let f(x)  4x  1 and g(x)  2x  5x
2
Find each composition g  f x 
Example 6 Finding Composition Functions
Let f(x)  4x  1 and g(x)  2x  5x
2
Find each composition f  g x 
Example 6 Finding Composition Functions
Let f(x)  4x  1 and g(x)  2x  5x
2
Find each composition f  g x 
Caution
In general,thecomposition functionf  g
is not thesame as theproduct fg.
For example,with f and g definedas
in Example6
f  g (x)  8x
2
 20x  1
But
fg(x)  4x  12 x
2

 5x  8x  22x  5x
3
2
Example 7 Finding Composite Functions and Their Domains
1
Let f(x) 
x
and g(x)  3 - x
Find thecomposition f  g x 
Give thedomainof f  gx 
Example 7 Finding Composite Functions and Their Domains
1
Let f(x) 
x
and g(x)  3 - x
Find thecomposition g  f x 
Give thedomainof g  f x 
Example 8 Finding Functions That Form a Given Composite
Find functionsf and g such that
 f  g x  ( x
2
 5)  4( x  5)  3
3
2