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A. The Composition of Functions

A. The Composition of Functions

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Section 3.6 The Composition of Functions and the Difference Quotient



EXAMPLE 1







353



Evaluating a Function

For g1x2 ϭ x2 Ϫ 3, find

a. g1Ϫ52

b. g15t2

c. g1t Ϫ 42



Solution







g1x2 ϭ x2 Ϫ 3



a.

input Ϫ5



g1Ϫ52 ϭ 1Ϫ52 2 Ϫ 3

ϭ 25 Ϫ 3

ϭ 22



simplify

result



g1x2 ϭ x Ϫ 3



b.



original function



g15t2 ϭ 15t2 2 Ϫ 3

ϭ 25t2 Ϫ 3



square input, then subtract 3

result



g1x2 ϭ x2 Ϫ 3



c.

input t Ϫ 4



It’s important to note that t and

t Ϫ 4 are two different, distinct

values—the number represented

by t, and a number four less than t.

Examples would be 7 and 3, 12

and 8, as well as Ϫ10 and Ϫ14.

There should be nothing awkward

or unusual about evaluating g(t)

versus evaluating g1t Ϫ 42 as in

Example 1(c).



square input, then subtract 3



2



input 5t



WORTHY OF NOTE



original function



original function



g1t Ϫ 42 ϭ 1t Ϫ 42 2 Ϫ 3

ϭ t2 Ϫ 8t ϩ 16 Ϫ 3

ϭ t2 Ϫ 8t ϩ 13



square input, then subtract 3

expand binomial

result



Now try Exercises 7 and 8







When the input value is itself a function (rather than a single number or variable),

this process is called the composition of functions. The evaluation method is exactly

the same, we are simply using a function input. Using a general function g(x) and a

function diagram as before, we illustrate the process in Figure 3.70.

Figure 3.70



Input x



g specifies

operations on x



g(x)

Input g(x)



Output

g(x)



f specifies

operations on

g(x)



f(x)



Output ( f Њ g)(x) = f [g(x)]



The notation used for the composition of f with g is an open dot “ ‫ ” ؠ‬placed between them, and is read, “f composed with g.” The notation 1 f ‫ ؠ‬g21x2 indicates that

g(x) is an input for f: 1 f ‫ ؠ‬g2 1x2 ϭ f 3g1x2 4 . If the order is reversed, as in 1g ‫ ؠ‬f 21x2, f 1x2

becomes the input for g: 1g ‫ ؠ‬f 21x2 ϭ g 3 f 1x2 4 . Figure 3.70 also helps us determine the

domain of a composite function, in that the first function g can operate only if x is a

valid input for g, and the second function f can operate only if g(x) is a valid input for f.

In other words, 1 f ‫ ؠ‬g2 1x2 is defined for all x in the domain of g, such that g(x) is in the

domain of f.

CAUTION







Try not to confuse the new “open dot” notation for the composition of functions, with the

multiplication dot used to indicate the product of two functions: 1f # g21x2 ϭ 1fg21x2 while

1f ‫ ؠ‬g21x2 ϭ f 3 g1x2 4 .



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The Composition of Functions

Given two functions f and g, the composition of f with g is defined by

1 f ‫ ؠ‬g2 1x2 ϭ f 3g1x2 4



The domain of the composition is all x in the domain of g

for which g(x) is in the domain of f.

In Figure 3.71, these ideas are displayed using mapping notation, as we consider

the simple case where g1x2 ϭ x and f 1x2 ϭ 1x.

Figure 3.71

fЊg



f



g



Domain of f Њ g

x2



f [g(x2)]

g(x1)



x1



Range of f Њ g



g(x2)



g

Range of g



Domain of f



Domain of g



Range of f



The domain of g (all real numbers) is shown within the red border, with g taking

the negative inputs represented by x1 (light red), to a like-colored portion of the

range—the negative outputs g(x1). The nonnegative inputs represented by x2 (dark

red) are also mapped to a like-colored portion of the range—the nonnegative outputs

g(x2). While the range of g is also all real numbers, function f can only use the nonnegative inputs represented by g(x2). This restricts the domain of 1 f ‫ ؠ‬g21x2 to only the

inputs from g, where g(x) is in the domain of f.

EXAMPLE 2







Finding a Composition of Functions



Given f 1x2 ϭ 1x Ϫ 4 and g1x2 ϭ 3x ϩ 2, find

a. 1 f ‫ ؠ‬g21x2

b. 1g ‫ ؠ‬f 21x2

Also determine the domain for each.



Solution







a. f 1x2 ϭ 1x Ϫ 4 says “decrease inputs by 4, and take the square root of the

result.”

1 f ‫ ؠ‬g21x2 ϭ f 3 g1x2 4

g (x ) is an input for f

ϭ 1g1x2 Ϫ 4

decrease input by 4, and take the square root of the result

ϭ 113x ϩ 22 Ϫ 4 substitute 3x ϩ 2 for g (x )

result

ϭ 13x Ϫ 2

While g is defined for all real numbers, f (x) represents a real number only

when x Ն 4. For f 3g1x2 4 , this means we need g1x2 Ն 4, giving 3x ϩ 2 Ն 4,

x Ն 23. In interval notation, the domain of 1 f ‫ ؠ‬g2 1x2 is x ʦ 3 23, q2 .



WORTHY OF NOTE



Example 2 shows that 1f ‫ ؠ‬g21x2 is

generally not equal to 1g ‫ ؠ‬f 21x2 . On

those occasions when they are

equal, the functions have a unique

relationship that we’ll study in

Section 5.1.



b. The function g says “inputs are multiplied by 3, then increased by 2.”

f (x ) is an input for g

1g ‫ ؠ‬f 21x2 ϭ g 3 f 1x2 4

multiply input by 3, then increase by 2

ϭ 3f 1x2 ϩ 2

ϭ 3 1x Ϫ 4 ϩ 2

substitute 1x Ϫ 4 for f (x )

For g[ f (x)], g can accept any real number input, but f can supply only those

where x Ն 4. The domain of 1g ‫ ؠ‬f 21x2 is x ʦ 34, q 2.

Now try Exercises 9 through 18







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Section 3.6 The Composition of Functions and the Difference Quotient



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Most graphing calculators have the ability to evaluate a composition of functions at

a given point. From Example 2, enter f(x) as Y1 ϭ 1x Ϫ 4 and g(x) as Y2 ϭ 3x ϩ 2.

To evaluate the composition 1 f ‫ ؠ‬g2 192 (since we know x ϭ 9 is in the domain), we can

(1) return to the home screen, enter Y1 1Y2 1922 and press

(Figure 3.72), or

(2) enter Y3 ϭ Y1 1Y2 1X2 2 on the Y= screen and use the TABLE feature (Figure 3.73).

See Exercises 19 through 24.

ENTER



Figure 3.72



EXAMPLE 3







Figure 3.73



Finding a Composition of Functions

For f 1x2 ϭ



3x

2

and g1x2 ϭ , analyze the domain of

x

xϪ1

a. 1 f ‫ ؠ‬g21x2

b. 1g ‫ ؠ‬f 2 1x2

c. Find the actual compositions and comment.



Solution







a. 1 f ‫ ؠ‬g21x2 : For g to be defined, x



0 is our first restriction. Once g(x) is used as

3g1x2

the input, we have f 3g1x2 4 ϭ

, and additionally note that g(x) cannot

g1x2 Ϫ 1

2

1, so x 2. The domain of f ‫ ؠ‬g is all real numbers

equal 1. This means

x

except x ϭ 0 and x ϭ 2.

b. 1g ‫ ؠ‬f 21x2 : For f to be defined, x 1 is our first restriction. Once f(x) is used as

2

the input, we have g 3f 1x2 4 ϭ

, and additionally note that f(x) cannot be 0.

f 1x2

3x

0, so x 0. The domain of 1g ‫ ؠ‬f 2 1x2 is all real numbers

This means

xϪ1

except x ϭ 0 and x ϭ 1.

c. For 1 f ‫ ؠ‬g21x2 :

3g1x2

f 3g1x2 4 ϭ

composition of f with g

g1x2 Ϫ 1

3 2

a ba b

1 x

2

ϭ

substitute for g (x)

x

2

a bϪ1

x

6

x

x

6

simplify denominator; invert and multiply

ϭ

ϭ #

x

2Ϫx

2Ϫx

x

6

result

ϭ

2Ϫx



Notice the function rule for 1 f ‫ ؠ‬g21x2 has an implied domain of x 2, but does

not show that g (the inner function) is undefined when x ϭ 0 (see Part a). The

domain of 1 f ‫ ؠ‬g2 1x2 is actually all real numbers except x ϭ 0 and x ϭ 2.



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For 1g ‫ ؠ‬f 21x2 we have:

g 3f 1x2 4 ϭ



2

f 1x2

2

ϭ

3x

xϪ1

2 xϪ1

ϭ #

1

3x

21x Ϫ 12

ϭ

3x



composition of g with f



substitute



3x

for f (x)

xϪ1



invert and multiply



result



Similarly, the function rule for 1g ‫ ؠ‬f 21x2 has an implied domain of x 0, but

does not show that f (the inner function) is undefined when x ϭ 1 (see Part b).

The domain of 1g ‫ ؠ‬f 21x2 is actually all real numbers except x ϭ 0 and x ϭ 1.

Now try Exercises 25 through 30







As Example 3 illustrates, the domain of h1x2 ϭ 1 f ‫ ؠ‬g2 1x2 cannot simply be taken from

the new function rule for h. It must be determined from the functions composed to

obtain h. The graph of 1 f ‫ ؠ‬g21x2 is shown in Figure 3.74. We can easily see x ϭ 2 is not

in the domain as there a vertical asymptote at x ϭ 2. The fact that x ϭ 0 is also

excluded is obscured by the y-axis, but the table shown in Figure 3.75 confirms that

x ϭ 0 is likewise not in the domain.

Figure 3.74



Figure 3.75



10



Ϫ10



10



Ϫ10



To further explore concepts related to the domain of a composition, see Exercises 69 and 70.



Decomposing a Composite Function



WORTHY OF NOTE

The decomposition of a function is

not unique and can often be done

in many different ways.



Based on Figure 3.76, would you say that the circle is inside the

square or the square is inside the circle? The decomposition of a

composite function is related to a similar question, as we ask

ourselves what function (of the composition) is on the “inside”—

the input quantity — and what function is on the “outside.” For

instance, consider h1x2 ϭ 1x Ϫ 4, where we see that x Ϫ 4 is

“inside” the radical. Letting g1x2 ϭ x Ϫ 4 and f 1x2 ϭ 1x, we have

h1x2 ϭ 1 f ‫ ؠ‬g21x2 or f [g(x)].



Figure 3.76



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Section 3.6 The Composition of Functions and the Difference Quotient



EXAMPLE 4







Decomposing a Composite Function



Solution







3

Noting that 1

x ϩ 1 is inside the squaring function, we assign g(x) as this inner

3

function: g1x2 ϭ 1 x ϩ 1. The outer function is the squaring function decreased

by 3, so f 1x2 ϭ x2 Ϫ 3.



A. You’ve just seen how we

can compose two functions

and determine the domain, and

decompose a function



3

Given h1x2 ϭ 1 1 x ϩ 12 2 Ϫ 3, identify two functions f and g so that

1 f ‫ ؠ‬g21x2 ϭ h1x2, then check by composing the functions to obtain h(x).



Check: 1 f ‫ ؠ‬g21x2 ϭ f 3g1x2 4

ϭ 3g1x2 4 2 Ϫ 3

3

ϭ 31

x ϩ 142 Ϫ 3

ϭ h1x2 ✓



g(x ) is an input for f

f squares inputs, then decreases the result by 3

substitute 13 x ϩ 1 for g (x)



Now try Exercises 31 through 34







B. A Numerical and Graphical View

of the Composition of Functions

Just as with the sum, difference, product, and quotient of functions, the composition of

functions can also be interpreted and understood graphically. For 1 f ‫ ؠ‬g21x2 , once the

value of g(x) is known (read from the graph), the value of 1 f ‫ ؠ‬g21x2 ϭ f 3g1x2 4 , can also

be determined.

EXAMPLE 5







Interpreting the Composition of Functions Numerically

For f(x) and g(x) as shown, use the graph given to

determine the value of each expression.

a. f(4), g(2), and 1 f ‫ ؠ‬g2 (2)

b. g(6), f(8), and 1g ‫ ؠ‬f 2 (8)

c. 1 f ‫ ؠ‬g2 (8) and 1g ‫ ؠ‬f 2 (0)

d. 1 f ‫ ؠ‬f 2 (9) and 1g ‫ ؠ‬g2 (0)



Solution







y

6



f (x)



Ϫ2



10 x



g(x)

Ϫ4



a. For f(4), we go to x ϭ 4 and note that 14, Ϫ12 is a

point on the red graph: f 142 ϭ Ϫ1. For g(2), go to x ϭ 2 and note that (2, 4)

is a point on the blue graph: g122 ϭ 4. Since g122 ϭ 4 and f 142 ϭ Ϫ1,

1 f ‫ ؠ‬g2122 ϭ f 3g122 4 ϭ f 142 ϭ Ϫ1: 1 f ‫ ؠ‬g2122 ϭ Ϫ1.

b. For g162, x ϭ 6 and we find that 16, Ϫ42 is a point on the blue graph: g162 ϭ Ϫ4.

For f 182, x ϭ 8 and note that that (8, 6) is a point on the red graph: f 182 ϭ 6. Since

f 182 ϭ 6 and g162 ϭ Ϫ4, 1g ‫ ؠ‬f 2 182 ϭ g3 f 182 4 ϭ g162 ϭ Ϫ4: 1g ‫ ؠ‬f 2182 ϭ Ϫ4.

c. As illustrated in parts (a) and (b), for 1f ‫ ؠ‬g2 182 ϭ f 3g182 4 we first determine

g(8), then substitute this value into f. From the blue graph g182 ϭ Ϫ2, and

f 1Ϫ22 ϭ 3. For 1g ‫ ؠ‬f 2102 ϭ g3 f 102 4 we first determine f(0), then substitute

this value into g. From the red graph f 102 ϭ Ϫ1, and g1Ϫ12 ϭ 1.

d. To evaluate 1 f ‫ ؠ‬f 2 192 ϭ f 3 f 192 4 we follow the same sequence as before. From

the graph of f, f 192 ϭ 5 and f 152 ϭ 1, showing 1 f ‫ ؠ‬f 2192 ϭ 1. The same ideas

are applied to 1g ‫ ؠ‬g2102 ϭ g 3g102 4 . Since g102 ϭ 4 and g142 ϭ Ϫ2,

1g ‫ ؠ‬g2102 ϭ Ϫ2.

Now try Exercises 35 and 36







As with other operations on functions, a composition can be understood graphically at specific points, as in Example 5, or for all allowable x-values. Exploring the

second option will help us understand certain transformations of functions, and why



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the domain of a composition is defined so carefully. For instance, consider the functions

f 1x2 ϭ Ϫx2 ϩ 3 and g1x2 ϭ 1x Ϫ 22 . The graph of f is a parabola opening downward

with vertex (0, 3). For h1x2 ϭ 1 f ‫ ؠ‬g2 1x2 we have h1x2 ϭ Ϫ1x Ϫ 22 2 ϩ 3, which

we recognize as the graph of f shifted 2 units to the right. Using Y1 ϭ ϪX2 ϩ 3,

Y2 ϭ X Ϫ 2, and Y3 ϭ Y1 1Y2 1X2 2 , we can deactivate Y2 so the only the graphs of Y1

and Y3 (in bold) are shown (see Figure 3.77). Here the domain of the composition was

not a concern, as both f and g have domain x ʦ ‫ޒ‬. But now consider f 1x2 ϭ 41x ϩ 1,

3

g1x2 ϭ

, and h1x2 ϭ 1 f ‫ ؠ‬g2 1x2 . The domain of g must exclude x ϭ 2 (which is

2Ϫx

also excluded from the domain of h), while the domain of f is x Ն Ϫ1. But after

3

assigning Y1 ϭ 4 2X ϩ 1, Y2 ϭ

, and Y3 ϭ Y1 1Y2 1X22 , the graph of Y3 shows

2ϪX

a noticeable gap between x ϭ 2 and x ϭ 5 (Figure 3.78). The reason is that

1 f ‫ ؠ‬g21x2 ϭ f 3g1x2 4 uses g(x) as the input for f, meaning for the domain, x Ն Ϫ1

3

Ն Ϫ1. Solving this inequality graphically (Figure 3.79)

becomes g1x2 Ն Ϫ1 S

2Ϫx

shows g1x2 Ն Ϫ1 only for the intervals x ʦ 1Ϫq, 22 ´ 3 5, q 2 , leaving the gap seen

in Y3 for the interval (2, 5]. The domain of h is x ʦ 1Ϫq, 22 ´ 3 5, q2 .

Figure 3.77

5



Ϫ5



5



Ϫ5



EXAMPLE 6



Figure 3.78



Figure 3.79



10



10



Ϫ5



10



Ϫ5







Ϫ5



10



Ϫ5



Interpreting a Composition Graphically and Understanding the Domain

Given f 1x2 ϭ 3 11 Ϫ x , g1x2 ϭ



4

,

xϩ3

a. State the domains of f and g.

b. Use a graphing calculator to study the graph of h1x2 ϭ 1 f ‫ ؠ‬g21x2 .

c. Algebraically determine the domain of h, and reconcile it with the graph.



Solution







a. The domain of g must exclude x ϭ Ϫ3, and for the domain of f we must have

x Յ 1.

4

b. For the graph of h, enter Y1 ϭ 3 21 Ϫ x, Y2 ϭ

, and Y3 ϭ Y1 1Y2 1X22 .

xϩ3

Graphing Y3 on the ZOOM 6:ZStandard screen shows a gap between x ϭ Ϫ3

and x ϭ 1 (Figure 3.80).

c. Since the composition 1 f ‫ ؠ‬g2 1x2 ϭ f 3g1x2 4 uses g(x) as the input for f,

4

x Յ 1 (the domain for f ) becomes g1x2 Յ 1 S

Յ 1. Figure 3.81

xϩ3

shows a graphical solution to this inequality, which indicates g1x2 Յ 1 for

x ʦ 1Ϫq, Ϫ32 ´ 3 1, q 2 . The domain of h is x ʦ 1Ϫq, Ϫ32 ´ 31, q2 .



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