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
#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:
Solutions:
a) $$g(f(n + 4))=8n + 11$$
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#2:
Solutions:
a) $$g(h(2n))=24n - 12$$
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#3:
Solutions:
a) $$f\left(g\left(\frac{x}{3}\right)\right)=\frac{-2x - 9}{3}$$
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#4:
Solutions:
a) $$f\left(g\left(\frac{a}{3}\right)\right)=\frac{a^3 + 3a^2 + 9}{9}$$
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#5:
Solutions:
a) $$g\left(f\left(\frac{n}{2}\right)\right)=\frac{n^3 - 40}{8}$$