Question

A company awards its salespeople 3% commission on sales below $20,000, 3.5% commission on sales from $20,000 to $30,00, and 4% commission for sales over $30,000. Select the piecewise function that shows the commission for x dollars in sales. (1 point) Responses f⎛⎝⎜x⎞⎠⎟=⎧⎩⎨⎪⎪0.03,0.035,0.04,0≤x<20,00020,000≤x≤30,000x>30,000⎞⎠⎟ f ( x ) = 0 . 03 , 0 ≤ x < 20 , 000 0 . 035 , 20 , 000 ≤ x ≤ 30 , 000 0 . 04 , x > 30 , 000 f⎛⎝⎜x⎞⎠⎟=⎧⎩⎨⎪⎪0.03x,0.035x,0.04x,0≤x<20,00020,000≤x≤30,000x>30,000⎞⎠⎟ f ( x ) = 0 . 03 x , 0 ≤ x < 20 , 000 0 . 035 x , 20 , 000 ≤ x ≤ 30 , 000 0 . 04 x , x > 30 , 000 f⎛⎝⎜x⎞⎠⎟=⎧⎩⎨⎪⎪0.03,0.035,0.04,0≤x≤20,00020,000<x<30,000x≥30,000⎞⎠⎟ f ( x ) = 0 . 03 , 0 ≤ x ≤ 20 , 000 0 . 035 , 20 , 000 < x < 30 , 000 0 . 04 , x ≥ 30 , 000 f⎛⎝⎜x⎞⎠⎟=⎧⎩⎨⎪⎪0.03x,0.035x,0.04x,0≤x≤20,00020,000<x<30,000x≥30,000⎞⎠⎟

Answer 1

To express the commission structure as a piecewise function \( f(x) \), we need to consider both the commission rates and the ranges of sales amounts correctly.

The commission rates are:

• 3% for sales below $20,000
• 3.5% for sales from $20,000 to $30,000
• 4% for sales over $30,000

The correct piecewise function representing the commission \( f(x) \) is:

\[ f(x) = \begin{cases} 0.03x & \text{if } 0 \leq x < 20000 \ 0.035x & \text{if } 20000 \leq x \leq 30000 \ 0.04x & \text{if } x > 30000 \end{cases} \]

This function accounts for the actual commission amount by multiplying the sales amount by the respective commission rate for each range.

Thus, the correct selection from the provided options is:

\[ f(x) = \begin{cases} 0.03x & 0 \leq x < 20000 \ 0.035x & 20000 \leq x \leq 30000 \ 0.04x & x > 30000 \end{cases} \]