 # 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 ka1leE 2020-11-08 Answered

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

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Step 1 Let
a) $f\left(x\right)=x-400$
Thus, f models the price of a computer after a discount of 400 dollars.
Also, $g\left(x\right)=0.75x$
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 $\left(f\circ g\right)\left(x\right),$
$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)=f\left(0.75x\right)$
By using $f\left(x\right)=x-400$,
$f\left(g\left(x\right)\right)=0.75x-400$.
thus, the composite function $\left(f\circ g\right)\left(x\right)=0.75x-400.$
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 $\left(g\circ f\right)\left(x\right),$
$\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x-400\right)$
By using $g\left(x\right)=0.75x$ ,
$g\left(f\left(x\right)\right)=0.75\left(x-400\right)$.
thus, the composite function $\left(g\circ f\right)\left(x\right)=0.75\left(x-400\right)$
The composite function g(f(x)) models that the price of a computer has 400 dollars discount and then 25% discount.
d) Now find $\left(f\circ g\right)\left(x\right)-\left(g\circ f\right)\left(x\right),$
$\left(0.75x-400\right)-\left(0.75\left(x-400\right)\right)$
$=0.75x-400-\left(0.75x-300\right)$
$=0.75x-400+0.75x+300$
$=-100$
Thus, $\left(f\circ g\right)\left(x\right)-\left(g\circ f\right)\left(x\right)<0$
Hence $\left(f\circ g\right)\left(x\right)<\left(g\circ f\right)\left(x\right)$
Hence the price of computer using function $\left(f\circ g\right)\left(x\right)$ is minimum.
Thus the composite function $\left(f\circ g\right)\left(x\right)$ gives more discount than $\left(g\circ f\right)\left(x\right)$ .