About Function Composition:

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