Question

You make a purchase at a local hardware store, but what you've bought is too big to take home in your car. For a small fee, you arrange to have the hardware store deliver your purchase for you. You pay for your purchase, plus the sales taxes, plus the fee. The taxes are 7.5% and the fee is $20. (i) Write a function t(x) for the total, after taxes, on the purchase amount x. Write another function f(x) for the total, including the delivery fee, on the purchase amount x. (ii) Calculate and interpret (f o t)(x) and (t o f )(x). Which results in a lower cost to you

Answer 1

Answer:

$f(x) = 1.075x$

$t(x) = x + 20$

$(f\ o\ t)(x) = 1.075x + 21.5$

$(t\ o\ f)(x) = 1.075x + 20$

Explanation:

Given

$Tax = 7.5\%$

$Fee = \ 20$ -- delivery

Solving (a): The function for total cost, after tax.

This is calculated as:

$f(x) = Tax *(1 + x)$

Where:

$x \to$ total purchase

So, we have:

$f(x) = x * (1 + 7.5\%)$

$f(x) = x * (1 + 0.075)$

$f(x) = x * 1.075$

$f(x) = 1.075x$

Solving (b): Include the delivery fee

$t(x) = x + Fee$

$t(x) = x + 20$

Solving (c): (f o t)(x) and (t o f)(x)

$(f\ o\ t)(x) = f(t(x))$

We have:

$f(x) = 1.075x$

So:

$f(t(x)) = 1.075t(x)$

This gives:

$f(t(x)) = 1.075*(x + 20)$

Expand

$f(t(x)) = 1.075x + 21.5$

So:

$(f\ o\ t)(x) = 1.075x + 21.5$

$(t\ o\ f)(x) = t(f(x))$

We have:

$t(x) = x + 20$

So:

$t(f(x)) = f(x) + 20$

This gives:

$t(f(x)) = 1.075x + 20$

We have:

$(f\ o\ t)(x) = 1.075x + 21.5$ ---- This represents the function to pay tax on the item and on the delivery

$(t\ o\ f)(x) = 1.075x + 20$ --- This represents the function to pay tax on the item only

The x coefficients in both equations are equal.

So, we compare the constants

$20 < 21.5$ means that (t o f)(x) has a lower cost