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Quizzes > Mathematics > Algebra

Ready to Ace the Composition of Functions Quiz?

Sharpen your composite function skills with this quick quiz!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art puzzle shapes and function symbols on dark blue background for composite functions quiz

This composition of functions quiz helps you practice building and evaluating f(g(x)) and g(f(x)) so you get faster and avoid common slips. Use it to spot gaps before a test and get comfortable with notation. Want more practice? Try a similar quiz , or brush up first with a quick algebra review.

Given f(x) = 2x + 1 and g(x) = x + 3, what is (f ? g)(4)?
17
15
7
9
To compute (f ? g)(4), first evaluate g(4) = 4 + 3 = 7, then f(7) = 2·7 + 1 = 15. Composition means you apply g first, then f. This verifies the correct answer is 15. For more on function composition, see Math is Fun: Function Composition.
If f(x) = x² and g(x) = ?x, what is (f ? g)(9)?
6
3
9
81
Compute g(9) = ?9 = 3, then f(3) = 3² = 9. The composite f(g(x)) applies the square root first, then squares it. Hence the result is 9. For more details, visit Math is Fun: Function Composition.
Given f(x) = 3x and g(x) = x - 5, find (g ? f)(2).
6
1
10
-3
First compute f(2) = 3·2 = 6, then g(6) = 6 - 5 = 1. Composition g(f(x)) means apply f then g. The result is therefore 1. See Khan Academy: Function Composition.
Is the composition of functions commutative, meaning does (f ? g)(x) always equal (g ? f)(x)?
Only if the functions are inverses
Never
Sometimes
Always
In general, f ? g and g ? f are not equal, but they can be equal for certain pairs of functions that commute. Thus composition is sometimes commutative. For more insight, see .
Let f(x) = 1/x and g(x) = 1/(x - 2). What is (f ? g)(x) simplified?
x/(x - 2)
2 - x
x - 2
1/[x(x - 2)]
Compute g(x) = 1/(x - 2), then f(g(x)) = 1 ÷ [1/(x - 2)] = x - 2. The reciprocal of the reciprocal returns the original denominator. For more, see Purplemath: Function Composition.
Given f(x) = x² + 1 and g(x) = 3x - 4, express (f ? g)(x).
9x² - 24x + 17
9x² - 12x + 17
3x² - 3
6x² - 8x + 1
Compute g(x) = 3x - 4, then f(g(x)) = (3x - 4)² + 1 = 9x² - 24x + 16 + 1 = 9x² - 24x + 17. That expansion is key. See .
What is the domain of (f ? g)(x) if f(x) = ?x and g(x) = 2x - 5?
x ? 0
x ? 5
x ? 2.5
x > 2.5
For ?(2x - 5) to be real, the inside must be ? 0: 2x - 5 ? 0 ? x ? 2.5. Hence the composite's domain is x ? 2.5. For domain analysis, see Purplemath: Domain of Composite Functions.
If f(x) = ln(x) and g(x) = e^x, what is (g ? f)(5)?
5
1/5
ln(5)
e^5
Compute f(5) = ln(5), then g(ln(5)) = e^{ln(5)} = 5. The exponential and natural log are inverse operations. More at Khan Academy: Inverse Functions.
If f(x) = ax + b and g(x) = cx + d satisfy (f ? g)(x) = 6x + 7 and (g ? f)(x) = 6x + 9, what are a, b, c, and d?
a = 4, b = 3, c = 2, d = 1
a = 2, b = 3, c = 1, d = 4
a = 1, b = 2, c = 3, d = 4
a = 3, b = 4, c = 2, d = 1
We have f(g(x)) = a(cx + d) + b = acx + ad + b = 6x + 7 and g(f(x)) = c(ax + b) + d = acx + bc + d = 6x + 9. Equating coefficients gives ac = 6, ad + b = 7, bc + d = 9. Solving yields a=3, c=2, b=4, d=1. See .
Let f(u) = u³ and g(x) = x² + 1. What is the derivative of (f ? g)(x)?
6x²(x² + 1)
3(x² + 1)²
6(x² + 1)²
6x(x² + 1)²
By the chain rule, (f ? g)'(x) = f'(g(x))·g'(x) = 3[g(x)]²·(2x) = 6x(x² + 1)². Chain rule applications are fundamental in calculus. More at .
For f(x) = sin x and g(u) = u², what is (g ? f)'(?/2)?
-1
1
2
0
Compute derivative via chain rule: (g ? f)'(x) = 2·sin(x)·cos(x). At x = ?/2, sin(?/2) = 1 and cos(?/2) = 0, so 2·1·0 = 0. Read more at .
If f and g are inverse functions, what is (f ? g)(x)?
f(x) + g(x)
f(x)/g(x)
g(x) - f(x)
x
By definition of inverses, f(g(x)) = x for all x in the domain. This composition returns the identity function. For more on inverse functions, see Math is Fun: Inverse Functions.
Let f(x) = 1/(1 + e^(?x)) and g(y) = ln(y/(1 ? y)). What is (g ? f)(x)?
x
e^x
?x
1/(1 + e^(?x))
Since g is the logit function and f is the logistic function, they are inverses. Thus g(f(x)) = x. It cancels out exactly. See Wikipedia: Logit Function for more.
0
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Study Outcomes

  1. Understand Composite Function Concepts -

    Grasp the definition and notation of composition of functions to recognize how one function's output becomes another's input.

  2. Compute Function Compositions -

    Apply composite function practice techniques to calculate f ∘ g and g ∘ f for a variety of function types accurately.

  3. Identify Domain and Range Constraints -

    Analyze composition of functions problems to determine valid input values and resulting outputs, ensuring correct domain and range.

  4. Apply Composites to Solve Equations -

    Use your knowledge of composite functions quiz strategies to solve equations that involve nested functions step by step.

  5. Analyze Multi-Step Compositions -

    Break down complex, multi-layered composite functions into simpler parts for clearer problem-solving and error checking.

  6. Sharpen Problem-Solving Skills -

    Engage with composite functions quiz online questions to reinforce algebraic manipulation and boost your math confidence.

Cheat Sheet

  1. Definition of Composite Functions -

    Master that the composition f∘g(x)=f(g(x)) means you apply g first and then f to its result. For example, if g(x)=x² and f(x)=2x+3, then f∘g(x)=2(x²)+3=2x²+3. This basic formula is the foundation of any composition of functions quiz.

  2. Domain and Range Considerations -

    Always check that every x in the domain of g produces g(x) in the domain of f to avoid undefined expressions. For instance, if f(x)=√x and g(x)=x−1, then you need x−1≥0 so x≥1. Proper domain checks are a must-have skill for composite function practice.

  3. Order Matters: Non-Commutativity -

    Remember that f∘g usually differs from g∘f, so swapping functions changes the result. As an example, f(x)=x+1 and g(x)=x² give f(g(x))=x²+1 but g(f(x))=(x+1)². Treating order carefully will boost your confidence on any composite functions quiz online.

  4. Inside-Out Strategy -

    Use the "inside-out" approach: compute g(x) first, then plug into f to streamline solving complex problems. A handy mnemonic is "Do G Before F" to keep your steps organized. Consistent composite function practice with this method cuts errors and saves time.

  5. Inverse of a Composition -

    Know that (f∘g)❻¹ = g❻¹∘f❻¹, which flips the order of inverse operations. For example, if f(x)=2x and g(x)=x+3, then (f∘g)(x)=2x+6 and its inverse is (x/2)−3. This fact is crucial for advanced composition of functions problems and builds deeper function fluency.

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