The regular price of a computer is x dollars. Let f(x) = x - 400 and g(x) = 0.75x. Solve, a. Describe what the functions f and g model in terms of the price of the computer. b. Find (f @ g)(x) and describe what this models in terms of the price of the computer. c. Repeat part (b) for (g @ f)(x). d. Which composite function models the greater discount on the computer, f @ g or g @ f?

The regular price of a computer is x dollars. Let f(x) = x - 400 and g(x) = 0.75x. Solve, a. Describe what the functions f and g model in terms of the price of the computer. b. Find (f @ g)(x) and describe what this models in terms of the price of the computer. c. Repeat part (b) for (g @ f)(x). d. Which composite function models the greater discount on the computer, f @ g or g @ f?

Question
Composite functions
asked 2020-11-08
The regular price of a computer is x dollars. Let f(x) = x - 400 and g(x) = 0.75x. Solve,
a. Describe what the functions f and g model in terms of the price of the computer.
b. Find \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}\) and describe what this models in terms of the price of the computer.
c. Repeat part (b) for \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}\).
d. Which composite function models the greater discount on the computer, \(\displaystyle{f}\circ{g}\) or \(\displaystyle{g}\circ{f}\)?

Answers (1)

2020-11-09
Step 1 Let f(x) = x - 400 and g(x) = 0.75x
a) f(x) = x - 400
Thus, f models the price of a computer after a discount of 400 dollars.
Also, g(x) = 0.75x
The function g gives a value that is 075 into x that is 75% of x.
This means that g gives only 75% of the regular price.
Therefore, g models the price of a computer after a 25 % discount.
b) To find composite function \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)},\)
\(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}={f{{\left({0.75}{x}\right)}}}\)
By using f(x) = x -400,
f(g(x)) = 0.75x -400.
thus, the composite function \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}={0.75}{x}-{400}.\)
The composite function f(g(x)) models that the price of a computer has 25% discount and then 400 dollars discount.
c) To find composite function \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)},\)
\(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}={g{{\left({f{{\left({x}\right)}}}\right)}}}={g{{\left({x}-{400}\right)}}}\)
By using g(x) = 0.75x ,
g(f(x)) = 0.75(x -400).
thus, the composite function \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}={0.75}{\left({x}-{400}\right)}\)
The composite function g(f(x)) models that the price of a computer has 400 dollars discount and then 25% discount.
d) Now find \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}-{\left({g}\circ{f}\right)}{\left({x}\right)},\)
(0.75x-400) - (0.75(x-400))
= 0.75x-400 -(0.75x-300)
= 0.75x-400 + 0.75x+300
= -100
Thus, \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}-{\left({g}\circ{f}\right)}{\left({x}\right)}{<}{0}\)</span>
Hence \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}{<}{\left({g}\circ{f}\right)}{\left({x}\right)}\)</span>
Hence the price of computer using function \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}\) is minimum.
Thus the composite function \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}\) gives more discount than \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}\) .
0

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