Function Composition Test

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 Test:

#1:

Instructions: Find each value or expression.

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

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

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

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#2:

Instructions: Find each value or expression.

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

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

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

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#3:

Instructions: Find each value or expression.

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

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

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

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#4:

Instructions: Find each value or expression.

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

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

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

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#5:

Instructions: Find each value or expression.

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

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

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

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Written Solutions:

#1:

Solution:

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

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#2:

Solution:

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

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#3:

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

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#4:

Solution:

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

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#5:

Solution:

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

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