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 \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}\) .

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 \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}\) .