Question

Question: Dale works at a mattress store, and makes a salary plus commission. He has a $500 weekly salary and makes a 5% commission for sales over $3,000. Let ƒ(x) = .05x and g(x) = x − 3000. Write the composition (f o g)(x) and (g o f)(x). Which one represents Dale's commission?

Answer 1

(f o g)(x) represents Dale's commission

Explanation:

The composite function (f o g)(x) means g(x) is the domain of f

The composite function (g o f)(x) means h(x) is the domain of g

∵ f(x) = 0.05x

∵ g(x) = x - 3000

- Substitute x in f(x) by g(x) to find (f o g)(x)

∵ (f o g)(x) = f(x-3000)

∵ f(x-3000) = 0.05(x - 3000)

∴ f(x-3000) = 0.05x - 150

∴ (f o g)(x) = 0.05x - 150

- Substitute x in g(x) by f(x) to find (g o f)(x)

∵ (g o f)(x) = g(0.05x)

∵ g(0.05x) = 0.05x - 3000

∴ (g o f)(x) = 0.05x - 3000

∵ Dale's commission is 5% for sales over $3,000

∵ His sales is $x

- Subtract 3000 from x, then multiply the answer by 5%

∴ His commission = (x - 3000) × 5%

∵ 5% =
`(5)/(100)=0.05`

∴ His commission = (x - 3000) × 0.05

- Multiply each term in the bracket by 0.05

∴ His commission = 0.05(x) - 0.05(3000)

∴ His commission = 0.05x - 150

∵ (f o g)(x) = 0.05x - 150

∴ (f o g)(x) = Dali's commission

∴ (f o g)(x) represents Dale's commission