chapter 2.7
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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
y
x
Example 1 Stretching or Shrinking a GraphGraph each function
xxf
x |x|
-2-1012
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
0 )(
g
fd
Example 2 Using Operations on Functions and Determining Domains
xgfa )(
and 9 -8x f(x)Let 1-2x g(x)
Example 2 Using Operations on Functions and Determining Domains
xgfb )(
and 9 -8x f(x)Let 1-2x g(x)
Example 2 Using Operations on Functions and Determining Domains
xfgc )(
and 9 -8x f(x)Let 1-2x g(x)
Example 2 Using Operations on Functions and Determining Domains
xg
fd
)(
and 9 -8x f(x)Let 1-2x g(x)
Example 2 Using Operations on Functions and Determining Domains
and 9 -8x f(x)Let 1-2x g(x)
(e) Give the domains of the functions in parts
(a) – (d).
Example 3 Evaluating Combinations of Functions
4gf
If possible, use the given representations of functions f and g to evaluate
2 gf
1fg
0
g
f
Example 3 Evaluating Combinations of Functions
4gf
If possible, use the given representations of functions f and g to evaluate
2 gf
1fg
0
g
f
Example 3 Evaluating Combinations of Functions
4gf
If possible, use the given representations of functions f and g to evaluate
2 gf
1fg
0
g
f
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
xhx
xfhxfm
0,
h
h
xfhxf
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.
h
xfhxf
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)].
f.g written f, and g
functions ofn compositio thecalled is
g[f(x)] values-yith function w The
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.
get to
r A(r)in r for 5r ngsubstituti
byin timeoffunction a as expressed
becan area ther, radius of circle
a of area thegivesr A(r) Since
2
2
22 t25π5t trA
Example 5 Evaluating Composite Functions
1 -2x f(x)Let ,1x
4 g(x) and
2n compositioeach Find gf
Example 5 Evaluating Composite Functions
1 -2x f(x)Let ,1x
4 g(x) and
3n compositioeach Find fg
Example 5 Evaluating Composite Functions
1 -2x f(x)Let ,1x
4 g(x) and
fg ofdomain theFind
Example 6 Finding Composition Functions
1 4x f(x)Let x52x g(x) and 2
xfgn compositioeach Find
Example 6 Finding Composition Functions
1 4x f(x)Let x52x g(x) and 2
xgfn compositioeach Find
Example 6 Finding Composition Functions
1 4x f(x)Let x52x g(x) and 2
xgfn compositioeach Find
Caution
fg.product theas same not the is
gffunction n compositio thegeneral,In
1 20x 8x (x)gf
6 Examplein
as defined g and f with example,For
2
xx 5214x (x)fg
But2 xx 522x8 23
Example 7 Finding Composite Functions and Their Domains
x
1 f(x)Let x-3 g(x) and
xgfn compositio theFind
xgf ofdomain theGive
Example 7 Finding Composite Functions and Their Domains
x
1 f(x)Let x-3 g(x) and
xfgn compositio theFind
xfg ofdomain theGive
Example 8 Finding Functions That Form a Given Composite
such that g and f unctions Find f
3)5(4)5( 232 xxxgf
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