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AP Precalculus Flashcards: Composition Of Functions

Study Composition Of Functions in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Composition Of Functions

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QUESTION
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Find (g◯f)(x)(g \bigcirc f)(x) if f(x)=x+2f(x) = x + 2 and g(x)=3xg(x) = 3x.

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ANSWER

(g◯f)(x)=3(x+2)(g \bigcirc f)(x) = 3(x + 2). Apply gg to f(x)=x+2f(x) = x + 2: 3(x+2)3(x + 2).

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What this deck covers

This deck focuses on Composition Of Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Find (g◯f)(x)(g \bigcirc f)(x) if f(x)=x+2f(x) = x + 2 and g(x)=3xg(x) = 3x.

Answer: (g◯f)(x)=3(x+2)(g \bigcirc f)(x) = 3(x + 2). Apply gg to f(x)=x+2f(x) = x + 2: 3(x+2)3(x + 2).

Flashcard 2: What is the definition of the composition of functions?

Answer: It is applying one function to the results of another: (f◯g)(x)=f(g(x))(f \bigcirc g)(x) = f(g(x)). First apply g, then apply f to that result.

Flashcard 3: If f(x)=x−3f(x) = x - 3 and g(x)=x2g(x) = x^2, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=x2−3(f \bigcirc g)(x) = x^2 - 3. Apply ff to g(x)=x2g(x) = x^2: x2−3x^2 - 3.

Flashcard 4: Find (f◯g)(x)(f \bigcirc g)(x) if f(x)=x2+1f(x) = x^2 + 1 and g(x)=3xg(x) = 3x.

Answer: (f◯g)(x)=9x2+1(f \bigcirc g)(x) = 9x^2 + 1. Substitute g(x)=3xg(x) = 3x into ff: (3x)2+1(3x)^2 + 1.

Flashcard 5: What is (f◯g)(0)(f \bigcirc g)(0) for f(x)=2x+3f(x) = 2x + 3 and g(x)=x2g(x) = x^2?

Answer: (f◯g)(0)=3(f \bigcirc g)(0) = 3. g(0)=0g(0) = 0, then f(0)=3f(0) = 3.

Flashcard 6: If f(x)=x2f(x) = x^2 and g(x)=3x+1g(x) = 3x + 1, find (g◯f)(x)(g \bigcirc f)(x).

Answer: (g◯f)(x)=3x2+1(g \bigcirc f)(x) = 3x^2 + 1. Apply gg to f(x)=x2f(x) = x^2: g(x2)=3x2+1g(x^2) = 3x^2 + 1.

Flashcard 7: What is (g◯f)(x)(g \bigcirc f)(x) for f(x)=1xf(x) = \frac{1}{x} and g(x)=x2g(x) = x^2?

Answer: (g◯f)(x)=1x2(g \bigcirc f)(x) = \frac{1}{x^2}. Apply gg to f(x)=1xf(x) = \frac{1}{x}: (1x)2(\frac{1}{x})^2.

Flashcard 8: If f(x)=4xf(x) = 4x and g(x)=x−2g(x) = x - 2, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=4(x−2)(f \bigcirc g)(x) = 4(x - 2). Apply ff to g(x)=x−2g(x) = x - 2: 4(x−2)4(x - 2).

Flashcard 9: Write (f◯g)(x)(f \bigcirc g)(x) for f(x)=x3f(x) = x^3 and g(x)=1xg(x) = \frac{1}{x}.

Answer: (f◯g)(x)=(1/x)3(f \bigcirc g)(x) = (1/x)^3. Substitute g(x)=1xg(x) = \frac{1}{x} into f(x)=x3f(x) = x^3.

Flashcard 10: If f(x)=x2f(x) = x^2 and g(x)=x+1g(x) = x + 1, what is (f◯g)(1)(f \bigcirc g)(1)?

Answer: (f◯g)(1)=4(f \bigcirc g)(1) = 4. g(1)=2g(1) = 2, then f(2)=4f(2) = 4.

Flashcard 11: Determine (f◯g)(x)(f \bigcirc g)(x) for f(x)=x2f(x) = x^2 and g(x)=x+1g(x) = x + 1.

Answer: (f◯g)(x)=(x+1)2(f \bigcirc g)(x) = (x + 1)^2. Apply ff to g(x)=x+1g(x) = x + 1: (x+1)2(x + 1)^2.

Flashcard 12: If f(x)=5xf(x) = 5x and g(x)=x−3g(x) = x - 3, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=5(x−3)(f \bigcirc g)(x) = 5(x - 3). Substitute g(x)=x−3g(x) = x - 3 into f(x)=5xf(x) = 5x.

Flashcard 13: Determine (f◯g)(1)(f \bigcirc g)(1) if f(x)=x−2f(x) = x - 2 and g(x)=x2g(x) = x^2.

Answer: (f◯g)(1)=−1(f \bigcirc g)(1) = -1. g(1)=1g(1) = 1, then f(1)=−1f(1) = -1.

Flashcard 14: What is (f◯g)(x)(f \bigcirc g)(x) for f(x)=x3f(x) = x^3 and g(x)=1xg(x) = \frac{1}{x}?

Answer: (f◯g)(x)=1x3(f \bigcirc g)(x) = \frac{1}{x^3}. Apply ff to g(x)=1xg(x) = \frac{1}{x}: (1x)3(\frac{1}{x})^3.

Flashcard 15: If f(x)=1xf(x) = \frac{1}{x} and g(x)=x+2g(x) = x + 2, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=1x+2(f \bigcirc g)(x) = \frac{1}{x + 2}. Apply ff to g(x)=x+2g(x) = x + 2: 1x+2\frac{1}{x + 2}.

Flashcard 16: What is (g◯f)(5)(g \bigcirc f)(5) for f(x)=3xf(x) = 3x and g(x)=x−2g(x) = x - 2?

Answer: (g◯f)(5)=13(g \bigcirc f)(5) = 13. f(5)=15f(5) = 15, then g(15)=13g(15) = 13.

Flashcard 17: If f(x)=x2f(x) = x^2 and g(x)=x+2g(x) = x + 2, find (g◯f)(x)(g \bigcirc f)(x).

Answer: (g◯f)(x)=x2+2(g \bigcirc f)(x) = x^2 + 2. Apply gg to f(x)=x2f(x) = x^2: x2+2x^2 + 2.

Flashcard 18: Find (f◯g)(−1)(f \bigcirc g)(-1) if f(x)=x2f(x) = x^2 and g(x)=x+5g(x) = x + 5.

Answer: (f◯g)(−1)=16(f \bigcirc g)(-1) = 16. g(−1)=4g(-1) = 4, then f(4)=16f(4) = 16.

Flashcard 19: If f(x)=x2+2f(x) = x^2 + 2 and g(x)=4xg(x) = 4x, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=16x2+2(f \bigcirc g)(x) = 16x^2 + 2. Substitute g(x)=4xg(x) = 4x into ff: (4x)2+2(4x)^2 + 2.

Flashcard 20: Calculate (g◯f)(3)(g \bigcirc f)(3) for f(x)=2xf(x) = 2x and g(x)=x2g(x) = x^2.

Answer: (g◯f)(3)=36(g \bigcirc f)(3) = 36. f(3)=6f(3) = 6, then g(6)=36g(6) = 36.

Flashcard 21: What is (f◯g)(x)(f \bigcirc g)(x) for f(x)=12xf(x) = \frac{1}{2}x and g(x)=x+4g(x) = x + 4?

Answer: (f◯g)(x)=12(x+4)(f \bigcirc g)(x) = \frac{1}{2}(x + 4). Apply ff to g(x)=x+4g(x) = x + 4.

Flashcard 22: Calculate (g◯f)(−3)(g \bigcirc f)(-3) for f(x)=3x+1f(x) = 3x + 1 and g(x)=x2g(x) = x^2.

Answer: (g◯f)(−3)=64(g \bigcirc f)(-3) = 64. f(−3)=−8f(-3) = -8, then g(−8)=64g(-8) = 64.

Flashcard 23: Find (g◯f)(0)(g \bigcirc f)(0) for f(x)=x2+1f(x) = x^2 + 1 and g(x)=2xg(x) = 2x.

Answer: (g◯f)(0)=2(g \bigcirc f)(0) = 2. f(0)=1f(0) = 1, then g(1)=2g(1) = 2.

Flashcard 24: Determine (f◯g)(x)(f \bigcirc g)(x) if f(x)=1/xf(x) = 1/x and g(x)=x−5g(x) = x - 5.

Answer: (f◯g)(x)=1/(x−5)(f \bigcirc g)(x) = 1/(x - 5). Apply ff to the result g(x)=x−5g(x) = x - 5.

Flashcard 25: State the notation for composing functions f(x)f(x) and g(x)g(x).

Answer: (f◯g)(x)=f(g(x))(f \bigcirc g)(x) = f(g(x)). Standard notation for function composition.

Flashcard 26: If f(x)=2x+3f(x) = 2x + 3, find (f◯f)(x)(f \bigcirc f)(x).

Answer: (f◯f)(x)=2(2x+3)+3(f \bigcirc f)(x) = 2(2x + 3) + 3. Substitute f(x)f(x) into itself.

Flashcard 27: If f(x)=x−4f(x) = x - 4 and g(x)=x2g(x) = x^2, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=x2−4(f \bigcirc g)(x) = x^2 - 4. Apply ff to g(x)=x2g(x) = x^2: x2−4x^2 - 4.

Flashcard 28: Evaluate (g◯f)(2)(g \bigcirc f)(2) if f(x)=x+1f(x) = x + 1 and g(x)=2xg(x) = 2x.

Answer: (g◯f)(2)=6(g \bigcirc f)(2) = 6. f(2)=3f(2) = 3, then g(3)=6g(3) = 6.

Flashcard 29: If f(x)=2xf(x) = 2x and g(x)=x+3g(x) = x + 3, what is (f◯g)(x)(f \bigcirc g)(x)?

Answer: (f◯g)(x)=2(x+3)(f \bigcirc g)(x) = 2(x + 3). Substitute g(x)=x+3g(x) = x + 3 into ff.

Flashcard 30: Find (g◯f)(x)(g \bigcirc f)(x) if f(x)=x2f(x) = x^2 and g(x)=1xg(x) = \frac{1}{x}.

Answer: (g◯f)(x)=1x2(g \bigcirc f)(x) = \frac{1}{x^2}. Apply gg to f(x)=x2f(x) = x^2: 1x2\frac{1}{x^2}.

Flashcard 31: If f(x)=2x+1f(x) = 2x + 1 and g(x)=x−3g(x) = x - 3, find (f◯g)(x)(f \bigcirc g)(x).

Answer: (f◯g)(x)=2(x−3)+1(f \bigcirc g)(x) = 2(x - 3) + 1. Substitute g(x)=x−3g(x) = x - 3 into ff.

Flashcard 32: If f(x)=x+1f(x) = x + 1 and g(x)=x2g(x) = x^2, what is (f◯g)(−2)(f \bigcirc g)(-2)?

Answer: (f◯g)(−2)=5(f \bigcirc g)(-2) = 5. g(−2)=4g(-2) = 4, then f(4)=5f(4) = 5.

Flashcard 33: What is (f◯g)(2)(f \bigcirc g)(2) if f(x)=3xf(x) = 3x and g(x)=x−1g(x) = x - 1?

Answer: (f◯g)(2)=3(f \bigcirc g)(2) = 3. g(2)=1g(2) = 1, then f(1)=3f(1) = 3.

Flashcard 34: Find (f◯g)(x)(f \bigcirc g)(x) if f(x)=x+5f(x) = x + 5 and g(x)=x3g(x) = x^3.

Answer: (f◯g)(x)=x3+5(f \bigcirc g)(x) = x^3 + 5. Apply ff to g(x)=x3g(x) = x^3: x3+5x^3 + 5.

Flashcard 35: Calculate (f◯g)(x)(f \bigcirc g)(x) if f(x)=2x+1f(x) = 2x + 1 and g(x)=x3g(x) = x^3.

Answer: (f◯g)(x)=2x3+1(f \bigcirc g)(x) = 2x^3 + 1. Apply ff to g(x)=x3g(x) = x^3: 2x3+12x^3 + 1.

Flashcard 36: Determine (g◯f)(x)(g \bigcirc f)(x) if f(x)=x+1f(x) = x + 1 and g(x)=x3g(x) = x^3.

Answer: (g◯f)(x)=(x+1)3(g \bigcirc f)(x) = (x + 1)^3. Apply gg to f(x)=x+1f(x) = x + 1: (x+1)3(x + 1)^3.