Given that:
- The function that represents the amount of money he earns per room decorated:
[tex]f(x)=25x^2-100[/tex]
You know that "x" represents the number of rooms the interior designer designs.
- The function that represents the number of rooms he decorates:
[tex]g(x)=\frac{1}{5}x[/tex]
You know that "x" represents the number of hours he works.
Part A
You need to substitute the function g(x) into the first function and then simplify, in order to determine the amount of money the designer will make decorating a house as a function of the hours he works.
Then, you get:
[tex]f(g(x))=25(\frac{1}{5}x)^2-100[/tex][tex]f(g(x))=\frac{25\cdot1^2}{5^2}x^2-100[/tex][tex]f(g(x))=\frac{25}{25}x^2-100[/tex][tex]f(g(x))=x^2-100[/tex]
Part B
In order to determine the amount of money he will earn working 40 hours, you need to substitute this value into the function found in Part A, and then evaluate:
[tex]x=40[/tex]
You get:
[tex]f(g(40))=(40)^2-100[/tex][tex]f(g(40))=1500[/tex]
Part C
By definition the Difference Quotient Formula is:
[tex][/tex]
