Question

An interior designer wants to decorate a newly constructed house. The function f(c) - 49x^2 - 200 represents the amount of money he earns per room decorated, where xrepresents the number of rooms he designs. The function g(x) = 1/7x represents the number of rooms the Interior designer decorates, where x is the number of hours he works.Part A: Determine the amount of money the Interior designer will make decorating the house as a function of hours he works.Part B: If the newly constructed house requires 50 hours of work, how much will the interior designer earn? Show all necessary calculations. Part C: Determine an expression to represent the difference quotient for the function found in Part A. Show all necessary work.

Answer 1

Part A:

to find the amount of money the interior designer will make as a function of hours we need to find f(g(x)), so:

$\begin{gathered} g(x)=\frac{1}{7}x \\ f(g(x))=49(\frac{1}{7}x)^2-200 \\ f(g(x))=49(\frac{1}{49}x^2)-200 \\ f(g(x))=x^2-200 \end{gathered}$

so the answer is: x^2-200

Part B:

we can use the equation we just found, so x is the number of hours so x=50 and we need to find f(g(x))

$50^2-200=2300$

so the answer is: 2300