Composite Functions

Key Concepts:

Evaluate the compositions of functions.

Find compositions of functions.

Real World Application:

You work forty hours a week at a furniture store. You receive a $220 weekly salary, plus a 3% commision on sales over $5000. Assume that you sellenough this week to get the commission. Given the functions f (x) = 0.03x and g(x) = x – 5000, which of ( f o g)(x) and (g o f )(x) represents your commission?

• Well, ( f o g)(x) = f(g(x)) would mean that I would take my sales x, subtract off the $5000 that didn't get the commission, and then multiply by 3%. On the other hand, (g o f )(x) =
• g( f (x)) would mean that I would take my sales x, multiply by 3%, and then subtract $5000 from the result. This could land me in negative numbers! (Would I owe money to my boss?)
• So ( f o g)(x) does what we need it to do: ( f o g)(x) represents my commission.

SAT Prep Question:

In composite functions, the expressions are set up as two functions. What will f(g(x)) be when f(x)=2x^2−4 and g(x)=x+2?

• Let’s organize this information: f(g(x))=?
• f(x)=2x^2−4 g(x)=x+2
• Notice that g(x) is nested inside f(x). This indicates that g(x) is the input for x in the f function, which means that g(x) will replace every x in the function of f.
• Since g(x)=x+2, the question becomes: what is f(x+2)?
• x+2 is the input, therefore, it will replace every x in the equation of the function of f.
• We get f(x+2)=2(x+2)^2−4.
• This becomes the equation f(g(x))=2(x+2)^2−4.