Let f(x) = 1/x and g(x) = x^2 + 4x. What two numbers are not in the domain of fºg?
2 answers:
Answer:
0 and -4
Step-by-step explanation:
We have been given the following functions;
f(x) = 1/x
g(x) = x^2 + 4x
The first step would be to evaluate the composite function fºg.
fºg = f[g(x)]. We substitute g(x) in place of x in f(x)
fºg =
This is clearly a rational function which will be defined everywhere except where the expression in the denominator will be equal to 0.
We solve for x;
Therefore, the two numbers that are not in the domain of fºg are 0 and -4
For this case we have the following equations:
We must find (f_ {o} g) (x):
By definition of composition of functions we have to:
So:
We must find the domain of The domain will be given by the values for which the function is defined, that is, when the denominator is nonzero.
So, the roots are:
The domain is given by all real numbers except 0 and -4
Answer:
x other than 0 and -4
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We call:
T = Tom's paiting rate
G = George's paiting rate
M = Mario's paiting rate
As the first statement says that they can work together, then:
T + G + M = Painting rate working together, that is the surface area they paint per hour.
Since the problem says that they last four hours painting a
room, then we need to multiply the amount of hours times the painting rate that is equal the surface area of the room, so:
4(T + G + M) = 600
Applying a similar reasoning, the George and Mario's painting rate is:
0 + G + M
Note that 0 implies that Tom didn't work in this case.
As they last eight hours painting the room, then:
8(0 + G + M) = 600
The same reasoning is applied to Tom and George working together. In this case 0 implies that Mario didn't work, so:
6(T + G + 0) = 600
We have a system of linear equations in three variables T, G, M:
<span>4(T+G+M)=600
8(0+G+M)=600
6(T+G+0)=600
</span>
Solving:
(1) T + G + M = 150
(2) G + M = 75
(3) T + G = 100
Substitute (3) in (1):
100 + M = 150 ∴ M = 50
Replace M in (2):
G + 50 = 75 ∴ G = 25
Substitute G in (3)
T + 25 = 100 ∴ T = 75
Then, the painting rates are:
Tom = 75
George = 25
Mario = 50
<span>Solving for Time to determine how long it would take Tom to paint the room alone:
</span>
<span>Painted Area = Painting Rate×Time, so:
</span>
T = Tom's paiting rate
G = George's paiting rate
M = Mario's paiting rate
As the first statement says that they can work together, then:
T + G + M = Painting rate working together, that is the surface area they paint per hour.
Since the problem says that they last four hours painting a
4(T + G + M) = 600
Applying a similar reasoning, the George and Mario's painting rate is:
0 + G + M
Note that 0 implies that Tom didn't work in this case.
As they last eight hours painting the room, then:
8(0 + G + M) = 600
The same reasoning is applied to Tom and George working together. In this case 0 implies that Mario didn't work, so:
6(T + G + 0) = 600
We have a system of linear equations in three variables T, G, M:
<span>4(T+G+M)=600
8(0+G+M)=600
6(T+G+0)=600
</span>
Solving:
(1) T + G + M = 150
(2) G + M = 75
(3) T + G = 100
Substitute (3) in (1):
100 + M = 150 ∴ M = 50
Replace M in (2):
G + 50 = 75 ∴ G = 25
Substitute G in (3)
T + 25 = 100 ∴ T = 75
Then, the painting rates are:
Tom = 75
George = 25
Mario = 50
<span>Solving for Time to determine how long it would take Tom to paint the room alone:
</span>
<span>Painted Area = Painting Rate×Time, so:
</span>