1-6 function operations and composition of · pdf filefind ( f + g)(x), (f ± g)(x), (f...

Find (f + g )(x), (f g )(x), (f g )(x), and (x) for each f (x) and
g (x). State the domain of each new function.
1.f (x) = x2 + 4
g(x) =
SOLUTION:
D = [0, ) for all of the functions except , for which
.
2.f (x) = 8 x3 g(x) = x 3
SOLUTION:
D = ( , ) for all of the functions except , for which
.
3.f (x) = x2 + 5x + 6 g(x) = x + 2
SOLUTION:
.
4.f (x) = x 9 g(x) = x + 5
SOLUTION:
.
5.f (x) = x2 + x g(x) = 9x
SOLUTION:
. Even though there appears to be no restriction in
the simplified function, there is in the original.
6.f (x) = x 7 g(x) = x + 7
SOLUTION:
.
7.f (x) =
g(x) = x3 + x
SOLUTION:
for all functions. Even though there appears to be
no restriction in the simplified function (f g)(x), there is in the original.
8.f (x) =
g(x) =
SOLUTION:
for all of these functions. Even though there
appears to be no restriction in the simplified function (f g)(x) or
, there is in the originals.
9.f (x) =
g(x) = 4
SOLUTION:
D = (0, ) for all of these functions. Even though there appears to be no restriction in the simplified function (f g)(x), there is in the original.
10.f (x) =
g(x) = x4
SOLUTION:
for all of these functions. Even though there
appears to be no restriction in the simplified function (f g)(x), there is in the original.
11.f (x) =
g(x) = 3
SOLUTION:
For (f + g)(x), (f g)(x), and (f g)(x), D = [5, ). For ,
12.f (x) =
g(x) =
SOLUTION:
For (f + g)(x), (f g)(x), and (f g)(x), D = [4, ). For , D =
(4, ).
13.BUDGETING Suppose a budget in dollars for one person for one month is approximated by f (x) = 25x + 350 and g(x) = 15x + 200, where f is the cost of rent and groceries, g is the cost of gas and all other expenses, and x = 1 represents the total cost at the end of the firstweek. a. Find (f + g)(x) and the relevant domain. b. What (f + g)(x) represent? c. Find (f + g)(4). What does this value represent?
SOLUTION:a.
We are dealing with time, so the relevant domain is x0. b. (f + g)(x) all factors that influence the budget. c.
The budget for 4 weeks.
14.PHYSICS Two different forces act on the movement of an object being pushed across a floor: the force of the person pushing the object and the force of friction. If W is work in joules, F is force in newtons,
and d is displacement of the object in meters, Wp(d) = Fp d describes
the work of the person and Wf (d) = Ff d describes the work created by
friction. The increase in kinetic energy necessary to move the object is
the difference between the work done by the person Wp and the work
done by friction Wf .
a. Find (Wp Wf )(d).
b. Determine the net work expended when a person pushes a box 50 meters with a force of 95 newtons and friction exerts a force of 55 newtons.
SOLUTION:a.
b. Substitute d = 50, Fp = 95, and Ff = 55 into the expression d(Fp
Ff ).
The net work expended is 200 newtons per meter.
For each pair of functions, find [ f o g ](x), [g o f ](x), and [ f o g ](6).
15.f (x) = 2x 3 g(x) = 4x 8
SOLUTION:
16.f (x) = 2x2 5x + 1 g(x) = 5x + 6
SOLUTION:
17.f (x) = 8 x2
g(x) = x2 + x + 1
SOLUTION:
18.f (x) = x2 16
g(x) = x2 + 7x + 11
SOLUTION:
19.f (x) = 3 x2
g(x) = x3 + 1
SOLUTION:
20.f (x) = 2 + x4
g(x) = x2
SOLUTION:
Find f o g .
21.f (x) =
g(x) = x2 4
SOLUTION:
To find f g, you must first be able to find g(x) = x2 4, which can be done for all real numbers. Then you must be able to evaluate f (x) =
foreachoftheseg(x)-values, which can only be done when g(x)
1. This means that we must exclude from the domain those values
for which x2 4 = 1.
Therefore, the domain of f g is {x| x , x R}. Now find [f g](x).
Therefore, forx .
22.f (x) =
g(x) = x2 + 6
SOLUTION:
To find f g, you must first be able to find g(x) = x2 + 6, which can be
done for all real numbers. Then you must be able to evaluate f (x) =
foreachoftheseg(x)-values, which can only be done when g
(x) 3. This means that we must exclude from the domain those values
for which x2 + 6 = 3.
Since isnotarealnumber,therearenox-values in the domain of
g such that x2 + 6 = 3. This means that we do not need to restrict the
domain of f g; the domain of f g is all real numbers. Now find [f g](x).
Therefore, .
23.f (x) =
g(x) = x2 4
SOLUTION:
To find f g, you must first be able to find g(x) = x2 4, which can be
foreachoftheseg(x)-values, which can only be done when g(x) 4. This means that we must exclude from the domain those
values for which x2 4 < 4.
Since x2 will never be less than 0, we do not need to restrict the domain
of f g. The domain of f g is all real numbers. Now find [f g](x).
Therefore, .
24.f (x) = x2 9
g(x) =
SOLUTION:
To find f g, you must first be able to find g(x) = , which can
only be done for x 3. Then you must be able to evaluate f (x) = x2
9 for each of these g(x)-values, which can be done for all real numbers.
Therefore, the domain of f g is {x| x 3, x R}. Now find [f g](x).
Therefore forx 3.
25.f (x) =
g(x) =
SOLUTION:
To find f g, you must first be able to find g(x) = , which can
only be done for x 6. Then you must be able to evaluate f (x) = for
each of these g(x)-values, which can only be done when g(x) 0. This means that we must also exclude from the domain those values for
which = 0.
Therefore, x 6. Combining these restrictions, the domain of [f o g](x),is {x| x < 6, x R}. Now find [f g](x).
Therefore, forx < 6.
26.f (x) =
g(x) =
SOLUTION:
To find f g, you must first be able to find g(x) = , which can only be done for x 8. Then you must be able to evaluate f (x) =
foreachoftheseg(x)-values, which can only be done when g(x)
0. This means that we must also exclude from the domain those
values for which = 0.
Therefore, x 8. Combining these restrictions, the domain of [f o g](x), is {x| x > 8, x R}. Now find [f g](x).
Therefore, forx > 8.
27.f (x) =
g(x) = x2 + 4x 1
SOLUTION:
To find f g, you must first be able to find g(x) = x2 + 4x 1, which
can be done for all real numbers. Then you must be able to evaluate f
(x) = foreachoftheseg(x)-values, which can only be done when g(x) 5. This means that we must also exclude from the
domain those values for which x2 + 4x 1 < 5.
Since (x + 2)2 will never be less than 0, we do not need to restrict the
domain of f g. The domain of f g is all real numbers. Now find [f g](x).
Therefore, .
28.f (x) =
g(x) = x2 + 8
SOLUTION:
To find f g, you must first be able to find g(x) = x2 + 8, which can be
foreachoftheseg(x)-values, which can only be done when g(x) 2. This means that we must also exclude from the domain those
values for which x2 + 8 < 2.
Since x2 will never be less than 6, we do not need to restrict the
domain of f g. The domain of f g is all real numbers. Now find [f g](x).
Therefore, .
29.RELATIVITY In the theory of relativity, m(v) = , where c
is the speed of light, 300 million meters per second, and m is the mass ofa 100-kilogram object at speed v in meters per second. a. Are there any restrictions on the domain of the function? Explain their meaning. b. Find m(10), m(10,000), and m(1,000,000). c. Describe the behavior of m(v) as v approaches c. d. Decompose the function into two separate functions.
SOLUTION:a. {v|0v < c, v R}; The speed of the object v cannot be equal to
the speed of light c. Otherwise, we would obtain , which is
undefined. Also, the speed v cannot be greater than c. Otherwise, we would have to find the square root of a negative number, which is an imaginary number. Lastly, the speed cannot be less than zero because speed cannot be negative. b.
c. Test values closer to c.
As v approaches c, m(v)approaches.
d. Sample answer: Let and
Find two functions f and g such that h(x) = [f o g ](x). Neither function may be the identity function f (x) = x.
30.h(x) = + 7
SOLUTION:
Sample answer: f (x) = +7g(x) = 4x + 2
31.h(x) = 8
SOLUTION:
Sample answer: f (x) = 8; g(x) = x + 5
32.h(x) = | 4x + 8 | 9
SOLUTION:
Sample answer: f (x) = | x | 9; g(x) = 4x + 8
33.h(x) = [[3(x 9)]]
SOLUTION:Sample answer: f (x) = [[3x]]; g(x) = x 9
34.h(x) =
SOLUTION:
Sample answer: f (x) = ; g(x) =
35.
SOLUTION:
Sample answer: f (x) = x3; g(x) = +4
36.h(x) =
SOLUTION:
Sample answer: f (x) = ; g(x) = x + 2
37.h(x) =
SOLUTION:
Sample answer: f (x) = g(x) = x 5
38.h(x) =
SOLUTION:
Sample ans

1-6 function operations and composition of · pdf filefind ( f + g)(x), (f ± g)(x), (f...

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