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