Answers:
c(h) = 75h+100
s(c) = 0.0825c
t(h) = 1.0825*(75h+100)
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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.
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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.
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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%"
