When working with function composition, we are essentially plugging one function in as the input of another function. We then simplify and give our answer. When we see (f ○ g)(x) , f(g(x)), or f[g(x)], we are being asked to plug the function g(x) in for x in the function f(x).

Test Objectives
• Demonstrate a general understanding of function notation
• Demonstrate the ability to find the value of a function for a given input
• Demonstrate the ability to plug one function in as the input for another function and simplify
Function Composition Practice Test:

#1:

Instructions: Find each value or expression.

$$g(n)=4n - 1$$

$$f(n)=2n - 5$$

a) $$g(f(n + 4))$$

#2:

Instructions: Find each value or expression.

$$g(n)=3n - 3$$

$$h(n)=4n - 3$$

a) $$g(h(2n))$$

#3:

Instructions: Find each value or expression.

$$f(x)=x - 2$$

$$g(x)=-2x - 1$$

a) $$f\left(g\left(\frac{x}{3}\right)\right)$$

#4:

Instructions: Find each value or expression.

$$f(a)=3a + 1$$

$$g(a)=a^3 + a^2$$

a) $$f\left(g\left(\frac{a}{3}\right)\right)$$

#5:

Instructions: Find each value or expression.

$$g(n)=n - 1$$

$$f(n)=n^3 - 4$$

a) $$g\left(f\left(\frac{n}{2}\right)\right)$$

Written Solutions:

#1:

Solutions:

a) $$g(f(n + 4))=8n + 11$$

#2:

Solutions:

a) $$g(h(2n))=24n - 12$$

#3:

Solutions:

a) $$f\left(g\left(\frac{x}{3}\right)\right)=\frac{-2x - 9}{3}$$

#4:

Solutions:

a) $$f\left(g\left(\frac{a}{3}\right)\right)=\frac{a^3 + 3a^2 + 9}{9}$$

#5:

Solutions:

a) $$g\left(f\left(\frac{n}{2}\right)\right)=\frac{n^3 - 40}{8}$$