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 f(g(x)), f(g(2)), and f(g(x + 1)).

$$a)\hspace{.2em}$$ $$f(x)=2x - 1$$ $$g(x)=5x + 1$$

#2:

Instructions: find g(f(x)), g(f(-3)), and g(f(a - 1)).

$$a)\hspace{.2em}$$ $$f(x)=2x^2 - 5$$ $$g(x)=\frac{x}{7}$$

#3:

Instructions: find f(g(x)) and f(g(x2)).

$$a)\hspace{.2em}$$ $$f(x)=x^2 - 7x - 3$$ $$g(x)=\frac{2x}{3}$$

#4:

Instructions: find f(g(x)) and state the domain.

$$a)\hspace{.2em}$$ $$f(x)=x^2$$ $$g(x)=\sqrt{x - 1}$$

#5:

Instructions: find g(f(x)) and state the domain.

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x + 2}$$ $$g(x)=\frac{3}{x - 1}$$

#6:

Instructions: find f(x) and g(x).

$$a)\hspace{.2em}$$ $$f(g(x))=(x - 3)^2 - 5(x - 3) + 1$$

#7:

Instructions: find f(x) and g(x).

$$a)\hspace{.2em}$$ $$f(g(x))=\frac{1}{x^2 - 3x + 2}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ $$f(g(x))=10x + 1$$ $$f(g(2))=21$$ $$f(g(x + 1))=10x + 11$$

#2:

Solutions:

$$a)\hspace{.2em}$$ $$g(f(x))=\frac{2x^2 - 5}{7}$$ $$g(f(-3))=\frac{13}{7}$$ $$g(f(a - 1))=\frac{2a^2 - 4a - 3}{7}$$

#3:

Solutions:

$$a)\hspace{.2em}$$ $$f(g(x))=\frac{4x^2 - 42x - 27}{9}$$ $$f(g(x^2))=\frac{4x^4 - 42x^2 - 27}{9}$$

#4:

Solutions:

$$a)\hspace{.2em}$$ $$f(g(x))=x - 1$$ $$Domain: \{x | x ≥ 1 \}$$

#5:

Solutions:

$$a)\hspace{.2em}$$ $$g(f(x))=-\frac{3(x + 2)}{x + 1}$$ $$Domain: \{x | x ≠ -2, -1\}$$

#6:

Solutions:

$$a)\hspace{.2em}$$ $$f(x)=x^2 - 5x + 1$$ $$g(x)=x - 3$$

#7:

Solutions:

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=x^2 - 3x + 2$$