N needed for clarity. - Utah State University

108

Chapter 2 Functions

87. In Exercise 86, solve the problem if the ranger is 4 miles from P.

88. Looking Ahead to Calculus A solid has as its base the region in the xy-plane bounded by the circle x 2 y2 4. (a) If every vertical cross section perpendicular to the x-axis is a semicircle, express the area K of the cross section at a distance u from the origin as a function of u.

(b) Repeat part (a) if each vertical cross section is an isosceles triangle with an altitude half as long as its base (not a semicircle).

(c) Repeat part (a) if each vertical cross section is an equilateral triangle.

(d) Repeat part (a) if each vertical cross section is a rectangle whose base is twice its vertical height.

2.6 C O M B I N I N G F U N C T I O N S

What is proved about numbers will be a fact in any universe.

Julia Robinson

Just as we combine numbers to get other numbers, so we may combine functions to get other functions. The first four ways of combining functions give familiar sums, differences, products, or quotients, as we would expect. Composition, less familiar, is a key idea throughout much of what follows.

Definition: sum, difference, product, quotient functions

Suppose f and g are given functions. Functions denoted by f g, f g,

f

?

g,

and

f g

are

given

by:

Sum: f gx f x gx

Difference: f gx f x gx

Product: f ? gx f x ? gx

Quotient:

f g

x

f x gx

Neyman . . . interviewed

me [for a job at Berkeley and ] said he would let me know. . . . I didn't really expect anything to happen. I had already written 104 letters of application to black colleges. Eventually I got a letter from [Neyman] saying something like "In view of the war situation and the draft possibilities, they have decided to appoint a woman to this position." [My eventual appointment here came 12 years later.]

David Blackwell

The domain of each combined function is the set of all real numbers for which the right side of the equation is meaningful as a real number. Use parentheses as needed for clarity.

The definitions stated here are not mere formal manipulations of symbols. For instance, the plus sign in f g is part of the name of the function that assigns to each x the sum of two numbers, f x gx.

EXAMPLE 1 Combining functions If f x 4x 6 and gx 2x 2

3x, write an equation for

(a) f g and

(b)

f g

,

and

give

the

domain of each.

Solution

(a) f gx f x gx 4x 6 2x 2 3x 2x 2 7x 6.

The domain is the set of real numbers.

(b)

f g

x

f x gx

4x 2x2

6 3x

22x x2x

3 3 ,

which

simplifies

to

2 x

for

x

3 2

.

Therefore

gf x

2 x

,

where

the

domain

is

x

x

0

and

x

3 2

.

2.6 Combining Functions

109

Composition of Functions

Another way to combine functions is used frequently and plays an important role in both precalculus and calculus.

Definition: composition of functions

Suppose f and g are functions. The composition function, f g, read "f of g," is the function whose value at x is given by

f gx f gx. Thus to write a formula for f gx, in the rule defining f ,

replace each x in f~x! by g (x). The domain of f g is the set of all real numbers x such that both gx is defined, and f gx is defined.

y (0, 4)

(4, 0) ( f g)(x) = 4 ? ( x )2

= 4 ? x, x 0

(a) y (0, 2)

(? 2, 0)

(2, 0)

(g f )(x) = 4 ? x2

(b) FIGURE 37

The reason for calling the composition f g " f of g" is that the value of the composition function at a given number c is " f of gc."

EXAMPLE 2 Two compositions If f x 4 x 2 and gx x, (a) write an equation and (b) draw a calculator graph of (i) f g (ii) g f.

x Solution

(a) For each composition, we follow the procedure given in the definition. (i) f gx f gx 4 gx2 4 x2. For x to be a real number we must have x 0, and when x 0, we can simplify the equation for f g:

f gx 4 x, where x 0.

(ii) g f x g f x f x 4 x 2. Again, the domain is limited: for 4 x 2 0, we have 2 x 2.

(b) With a graphing calculator we can always enter the compositions in the form we wrote above, Y1 4 (X)2 and Y2 (4 X2).

If your calculator has a Y menu where you can enter several functions,

x

there are other options. For example, having entered f and g as Y1 4 X2 and

Y2 X, since f ( gx 4 gx2, we can enter f g as Y3 4 Y22 and g f as Y4 Y1. Observe that for f g we follow the defining rule for composition functions: replace each x in f x by gx.

The calculator graphs are shown in Figure 37. Note that the limitations on

the domain are obvious from the graphs and that we can also read off the

ranges. The range of f g is (, 4, and the range of g f is the closed interval 0, 2.

Alternate Solution Sometimes it is easier to verbalize the rules that define functions. The rules for f and g state that, for any given input, f squares the input and subtracts the result from 4, while g takes the square root of its input. Thus, suppose

x is the input. The function f squares x and subtracts the result from 4: 4 x2. Similarly, when g is applied to f x, g takes the square root of f x. The output is: g f x f x 4 x 2.

110

Chapter 2 Functions

Example 2 shows that f g and g f are not the same function. In general f g and g f are different, although there are important exceptions, as the next example demonstrates.

EXAMPLE 3 Equal compositions If f x 3x 8 and gx x 8 , 3

write an equation that gives the rule of correspondence for (a) f g (b) g f.

Solution Here the rule for f is "triple the input and then subtract 8;" for g it is "add 8 to the input and then divide the sum by 3."

(a)

f gx f gx f

x8 3

3 x8 3

8 x.

(b)

g

f x

g f x

g3x

8

3x

8 3

8

x.

Thus f gx g f x for every number x. We say that the two functions f g and g f are equal, f g g f.

Strategy: Write each equation in a more familiar form.

EXAMPLE 4 Composition equations If f x x 2 2x and gx 3 x, solve the equations.

(a) f gx 0 Solution

(b) g f x x 2 5 0

(a) f gx f gx f 3 x 3 x2 23 x x 2 4x 3. Thus the given equation becomes

x2 4x 3 0 or x 1x 3 0.

The solutions are 1 and 3. (b) g f x g f x gx 2 2x 3 x 2 2xx 2 2x 3. Re-

placing g f x by x 2 2x 3, the given equation becomes

x 2 2x 3 x 2 5 0 or 2x 8 0.

The solution is 4.

[? 5, 5] by [? 2.1, 4.1]

FIGURE 38 Fx g f x

EXAMPLE 5 Maxima and minima from calculator graphs the composition g f on the limited domain D 5, 5, where

f x

x2

2x

4x

4

5

and

gx x 2 3x.

Let F denote

(a) Draw a calculator graph of Fx g f x. (b) From your graph, find the maximum and minimum values of F. (c) Find the solution set for g f x 0.

Solution

(a) Writing a formula for the composition g f x requires us to replace each x in x 2 3x by the entire f x. The process is messy, to say the least, but some calculators are designed to make composition much easier. See the Technology

Tip following this example. Not knowing the range beforehand, we may set an x-range of 5, 5 to match the domain and adjust as necessary. A calculator graph is shown in Figure 38.

2.6 Combining Functions

111

(b) Using the TRACE function on the graph of y Fx, we find the low point near 1, 2 and the high point near 3, 4. Shifting a decimal window, we confirm that the the maximum value of F is 4 and the minimum value is 2.

(c) The graph crosses the x-axis at 2, 0, as is easily verified by evaluating F2,

and is above the x-axis for all values of x (from the domain of F) greater than 2. Thus the solution set for g f x 0 is the interval 2, 5.

TECHNOLOGY TIP Graphing compositions and defining functions

When composing functions, let the calculator do the hard work. In Example 5, to enter g f x f x2 3 f x, we need lots of parentheses:

Y ((2X 4)(X2 4X 5))2 3((2X 4)(X2 4X 5)).

If your calculator allows you to enter a list of functions, Y1, Y2, . . . , (TI and Casio)

then you can enter the composition function much more simply. First enter f as Y1 and then use Y1 to enter g f x as Y2 gY1 :

Y1 2X 4X2 4X 5 Y2 Y12 3Y1.

HP?38 HP? 48

Having entered functions F1X 2 * X 4X2 4 * X 5 and F2X X2 3 * X, write the composition as F3(X) F2(F1(X)) and graph.

On the home screen, enter each function as an equation, F(X) 2 * X 4X2 4 * X 5. Then press the DEF key (above STO). Similarly for GX X2 3 * X. Then on the PLOT screen enter the function as G(F(X)) and graph.

Strategy: Get simpler expressions for the composition function, substitute, and solve.

EXAMPLE 6 Composition inequality If f x x 2 9 and gx 2x 5, find the solution for f gx 0.

Solution f gx f 2x 5 2x 52 9. Therefore the given inequality may be written as 2x 52 9 0. This is equivalent to

2x 52 9 or 3 2x 5 3 or 1 x 4.

The solution set is x 1 x 4.

y

(0, 4)

y = x2 ? 5x + 4

(1, 0)

(4, 0)

x

(3, ? 2) y = (2x ? 5)2 ? 9

Alternate Solution Graphical We have seen often that calculator graphs allow us to read the solution set for an inequality such as 2x 52 9 0 or 4x 2 20x 16 0. We can graph Y 2X 52 9 or we can simplify the inequality to an equivalent form, by dividing through by 4, getting x 2 5x 4 0, and graph Y X2 5X 4. In either case we have a parabola that crosses the x-axis at 1, 0 and 4, 0. See Figure 39. We read the solution set as x 1 x 4.

EXAMPLE 7 Applied composition An oil spill on a lake assumes a circular shape with an expanding radius r given by r t 1, where t is the number of minutes after measurements are started and r is measured in meters.

(3, ? 8) FIGURE 39

(a) Find a formula that gives the area A of the circular region at any time t. (b) What is the area at the beginning measurement t 0? What is the area 3

minutes later?

112

Chapter 2 Functions

Strategy: (a) Since

r t 1 is a function of t and A r 2 is a function of r, then by composing functions we can express A as a function of t.

Solution

(a) Follow the strategy. A t 12 t 1.

Thus A as a function of t is A t .

When t is 0, A ? 0 square meters. When t is 3, A 3 4 square meters.

Calculator Evaluations

Many function evaluations by calculator actually involve composition of functions, especially with calculators that use "Reverse Polish" operations. With such a calculator, to evaluate Fx x 2 1 when x is 3, we enter 3, square it, and add 1, after which we take the square root. This amounts to treating F as a composition f g, where gx x 2 1 and f x x. We accomplish the same thing if we have a graphing calculator using an Algebraic Operating System when we use the ANS key. Using the same example, if we evaluate 32 1 and , ENTER the calculator displays 10. If we then evaluate , ANS we are taking the composition of the square root function with the previously evaluated x 2 1 function.

EXAMPLE 8

Function as a composition

If

Fx

x2

1

1

,

express

F

as

a

composition of two functions.

Solution

Let

f x

1 x

and

gx

x2

1.

Then

f gx

f x 2

1

x2

1

1

.

Thus, Fx is given by Fx f gx.

In problems of the type discussed in Example 8, be aware that there are many different solutions. For example, we could have taken

f x

x

1

1

and

gx x 2.

Then

f gx

f x 2

x2

1

1

.

Composition with Absolute Value

Composition of functions with the absolute value function affects graphs in a consistent fashion giving us two more useful basic transformations. It is easiest to look at a specific example.

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