Expert: Scotto - 10/4/2008

Question
Hey there. My professor gave us a huge 50 problem assignment and I am stuck on these few remaining. I would really appreciate your help!!

1. Consider the functions below:
f(x)=3rd root of x-1
g(x)=x^3 +1

Find each of the following, if possible. (If it is not possible, enter NONE.)
(a) f of g
(b) g of f
(c) (f of g)(0)

2. The number of bacteria in a certain food product is given by the model below, where T is the temperature of the food.
N(T) = 10T2 − 20T + 550,   1 ≤ T ≤ 20
When the food is removed from the refrigerator, the temperature of the food is given by T(t) = 4t + 1 where t is the time in hours.
(a) Find the composite function N(T(t)).

3. Consider the functions below:
f(x)=1/8x-9
g(x)=x^3

Use the functions above to find the function below.
g^-1 of f^-1

4. Find the constant of proportionality and write an equation that relates the variables.

q is inversely proportional to p, and q = 3 / 2 when p = 76.

5. f(x)= |x|
  g(x)= -x+4
Evaluate the following:
(a)  (f of g)(3)
(b)  (g of f)(2)

Thanks again!


Answer
1. Consider the functions f(x)=3rd√(x-1) g(x)=x^3 +1

Find each of the following, if possible. (If it is not possible, enter NONE.)
(a) f of g simplifies, for it is 3rd√(x^3), which is x.
(b) g of f simplifies, for it is 3rd√(x-1)^3+1, which is x-1+1 = x.
(c) (f of g)(0) is 0.


2. The number of bacteria in a certain food product is given by the model below, where T is the temperature of the food.
N(T) = 10T2 − 20T + 550,   1 ≤ T ≤ 20
When the food is removed from the refrigerator, the temperature of the food is given by T(t) = 4t + 1 where t is the time in hours.
(a) Find the composite function N(T(t)).
Is that 10T2 really 10T²?  
The answer is N(t) = 10(4t+1)² - 20(4t+1) + 550.
The first term can be sqaured {giving 10(16t²+8t+1) } and then like terms can be combined.

3. Consider the functions below:
f(x)=1/8x-9;  I will assume that the first part it is (1/8) x, but it could also be 1/(8x))
g(x)=x^3

Use the functions above to find the function below.
g^-1 of f^-1
As it is written, let f = x/8 - 9, so x/8 = f + 9, so x = 8f + 72.
For g, use the cuberoot on both sides.

4. Find the constant of proportionality and write an equation that relates the variables.

q is inversely proportional to p, and q = 3 / 2 when p = 76.

To be inversely proportional, q = a / p for some unknown constant a.
Put 76 in for p and 3/2 in for q, then solve to find a.

5. f(x)= |x|
 g(x)= -x+4
Evaluate the following:
(a)  (f of g)(3)
(b)  (g of f)(2)

a) Compute g(3), then put that into f.
b) Compute f(2), then put that into g.