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

Answer:

1.  $\( f\circ g(x)=0.05x-150$

2. $\( g\circ f(x)=0.05x-3000$

3. The first one represents Dale's commission

Explanation:

1. The composition of the function

                                                             $\( f\circ g(x)=f(g(x)) \)$    

means that you first apply the function g(x) and then f(x) on the output of g(x).

That is:

• f(x) = 0.05x
• g(x) = x - 3000

       $f(g(x)=0.05(x - 3000)$

       $f(g(x))=0.05x-150$

2. The composition of the function

                                                             $\( g\circ f(x)=g(f(x)) \)$                                                                  

means that you first apply the function f(x) and then g(x) on the output of f(x).

That is:

      $g(f(x))=((0.05x)-3000)=0.05x-3000$

3. Which one represents Dale's commission

To calculate Dales's commision you must subtract $3,000 from the sales, to find the sales over $3000. That is: x - 3,000, which is the function g(x).

Therefore, you first use g(x).

Then, you must multiply the output of g(x) by 0.05 to find the 5% of the sales over $3,000. That is: 0.05(g((x)) = 0.05(x - 3000) = 0.05x - 150.

Therefore, the composition that represents Dale's commission is the first one:

  $f(g(x)=0.05(x - 3000)$

       $f(g(x))=0.05x-150$