Chapter 6.4, Problem 7MCQ

(a).

To determine

Express the total price after the discount and the total price after the tax using function notation.

Answer to Problem 7MCQ

  $p(x)=x\cdot \left(1-\frac{25}{100}\right)=0.75x$

  $g(x)=x\cdot \left(1+\frac{6}{100}\right)=1.06x$

Explanation of Solution

Given:

Mrs. Ross is shopping for her children’s school chothes. She has a coupon for 25% off her total .The sales tax of 6% is added to the total after the coupon is applied.

Calculation:

Let x represent the price of the clothing ,

p(x) represent the price after the 25% discount :

So, $p(x)=x-25\%x=x\left(1-\frac{25}{100}\right)=0.75x$, and

g(x) represent the price after the tax 6% is added :

  $g(x)=x+6\%x=x\left(1+\frac{6}{100}\right)=1.06x$

(b).

To determine

Find composition of functions that represents the final price , $p\left[g(x)\right]\text{ or }g\left[p(x)\right]$ .

Answer to Problem 7MCQ

  $\left[g\circ p\right](x)$ represents the final price.

Explanation of Solution

Given:

Mrs. Ross is shopping for her children’s school chothes. She has a coupon for 25% off her total .The sales tax of 6% is added to the total after the coupon is applied.

Calculation:

From part (a) , we have:

The function for the price after discount :

  $p(x)=x\cdot \left(1-\frac{25}{100}\right)=0.75x$

The function for the price after sales tax :

  $g(x)=x\cdot \left(1+\frac{6}{100}\right)=1.06x$

Since, Ms.Ross pays sales tax on the discounted price of the clothes, we need to first apply the function that gives the price of clothes after the discount of 25% and then apply the function that applies the sales tax on the discounted price of the clothes.

Hence, the correct composition is :

  $\left[g\circ p\right](x)=g\left[p(x)\right]$ since it gives the output of price after the sales tax on the discounted price of clothes (input function of p(x)).

So,

  $\left[g\circ p\right](x)$ represents the final price of the clothes.