✨[ALGEBRA]✨ A plumbing company charges $75 per hour with a $100 inspection fee. A sales tax of 8.25% is then calculated based on the charges (subtotal). Write a function for the charges c(h) for the number of hours worked, h. Then write a function for the sales tax, s(c) collected on the charges. Finally, write a function for the total, t(h), using c(h) and the composite of s(h) and c(h).
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Answers:

c(h) = 75h+100

s(c) = 0.0825c

t(h) = 1.0825*(75h+100)

Explanation:

h = number of hours worked

75h = total cost if we ignore the taxes and inspection fee

75h+100 = total cost if we include inspection fee, but ignore taxes

c(h) = 75h+100 is the cost function before tax. It tells us what c will be equal to for any given value of h.

The s(c) function tells us what the sales tax s will be for any given cost c

Sales tax in this case is 8.25% which converts to the decimal form 0.0825

So we can say

s(c) = 0.0825c

We can replace each 'c' with c(h) to get

s( c(h) ) = 0.0825*c(h)

The reasoning for this is because we can then plug in the c(h) function we found earlier to get

s( c(h) ) = 0.0825*c(h)

s( c(h) ) = 0.0825*(75h+100)

s(h) = 0.0825*(75h+100)

This is the composite function your teacher is talking about. It's combing the two ideas in a sort of chain.

The total is going to be the cost before tax, plus the amount of sales tax

total cost = (charge before tax) + (sales tax amount)

t(h) = c(h) + s(h)

t(h) = (75h+100) + 0.0825*(75h+100)

t(h) = 1*(75h+100) + 0.0825*(75h+100)

t(h) = (1+0.0825)*(75h+100)

t(h) = 1.0825*(75h+100)

Note how the total cost is 1.0825 times of the cost before tax.

So we could say

t(h) = 1.0825*c(h)

The 1.0825 multiplier indicates "increase the value by 8.25%"

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