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Algebra 2 Flashcards: Using Intersections To Solve Equivalent Functions

Study Using Intersections To Solve Equivalent Functions in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

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What this deck covers

This deck focuses on Using Intersections To Solve Equivalent Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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.

Algebra 2 Flashcards: Using Intersections To Solve Equivalent Functions

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QUESTION

Find the intersection xxx-value: f(x)=3xf(x)=3^xf(x)=3x and g(x)=27g(x)=27g(x)=27.

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ANSWER

x=3x=3x=3. Set 3x=27=333^x=27=3^33x=27=33, so x=3x=3x=3.

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Flashcard 1: Find the intersection xxx-value: f(x)=3xf(x)=3^xf(x)=3x and g(x)=27g(x)=27g(x)=27.

Answer: x=3x=3x=3. Set 3x=27=333^x=27=3^33x=27=33, so x=3x=3x=3.

Flashcard 2: What is the intersection xxx-value of f(x)=x3f(x)=x^3f(x)=x3 and g(x)=0g(x)=0g(x)=0?

Answer: x=0x=0x=0. Set x3=0x^3=0x3=0, so x=0x=0x=0.

Flashcard 3: What is the intersection xxx-value if f(x)=g(x)f(x)=g(x)f(x)=g(x) occurs at point (4,−3)(4,-3)(4,−3)?

Answer: x=4x=4x=4. Intersection coordinates give the solution x-value directly.

Flashcard 4: Which midpoint does bisection test first on the interval [2,6][2,6][2,6]?

Answer: x=4x=4x=4. Bisection starts at the midpoint of the interval.

Flashcard 5: Which interval brackets a solution if h(1)=−2h(1)=-2h(1)=−2 and h(2)=3h(2)=3h(2)=3 for h(x)=f(x)−g(x)h(x)=f(x)-g(x)h(x)=f(x)−g(x)?

Answer: A solution lies in (1,2)(1,2)(1,2). Sign change from negative to positive indicates a zero crossing.

Flashcard 6: Find the intersection xxx-value(s): f(x)=x2f(x)=x^2f(x)=x2 and g(x)=−x2g(x)=-x^2g(x)=−x2.

Answer: x=0x=0x=0. Set x2=−x2x^2=-x^2x2=−x2: 2x2=02x^2=02x2=0, so x=0x=0x=0.

Flashcard 7: Find the intersection xxx-value(s): f(x)=x3f(x)=x^3f(x)=x3 and g(x)=xg(x)=xg(x)=x.

Answer: x=−1x=-1x=−1, x=0x=0x=0, and x=1x=1x=1. Set x3=xx^3=xx3=x and factor: x(x2−1)=x(x−1)(x+1)=0x(x^2-1)=x(x-1)(x+1)=0x(x2−1)=x(x−1)(x+1)=0.

Flashcard 8: Find the intersection xxx-value: f(x)=ln⁡(x)f(x)=\ln(x)f(x)=ln(x) and g(x)=ln⁡(5)g(x)=\ln(5)g(x)=ln(5).

Answer: x=5x=5x=5. Set ln⁡(x)=ln⁡(5)\ln(x)=\ln(5)ln(x)=ln(5): x=5x=5x=5.

Flashcard 9: Identify the intersection xxx-value: f(x)=xf(x)=xf(x)=x and g(x)=xg(x)=\sqrt{x}g(x)=x​.

Answer: x=0x=0x=0 and x=1x=1x=1. Set x=xx=\sqrt{x}x=x​ and square: x2=xx^2=xx2=x, so x(x−1)=0x(x-1)=0x(x−1)=0.

Flashcard 10: Identify the valid solution of ln⁡(x−2)=0\ln(x-2)=0ln(x−2)=0: x=2x=2x=2 or x=3x=3x=3?

Answer: x=3x=3x=3. Only x=3x=3x=3 makes the argument x−2=1>0x-2=1>0x−2=1>0 valid.

Flashcard 11: Find the intersection xxx-value: f(x)=ln⁡(x−2)f(x)=\ln(x-2)f(x)=ln(x−2) and g(x)=0g(x)=0g(x)=0.

Answer: x=3x=3x=3. Set ln⁡(x−2)=0\ln(x-2)=0ln(x−2)=0: x−2=1x-2=1x−2=1, so x=3x=3x=3.

Flashcard 12: Find the intersection xxx-value: f(x)=10xf(x)=10^xf(x)=10x and g(x)=1000g(x)=1000g(x)=1000.

Answer: x=3x=3x=3. Set 10x=1000=10310^x=1000=10^310x=1000=103, so x=3x=3x=3.

Flashcard 13: Find the intersection xxx-value: f(x)=exf(x)=e^xf(x)=ex and g(x)=1g(x)=1g(x)=1.

Answer: x=0x=0x=0. Set ex=1=e0e^x=1=e^0ex=1=e0, so x=0x=0x=0.

Flashcard 14: Find the intersection xxx-value: f(x)=ln⁡(x)f(x)=\ln(x)f(x)=ln(x) and g(x)=0g(x)=0g(x)=0.

Answer: x=1x=1x=1. Set ln⁡(x)=0\ln(x)=0ln(x)=0: x=e0=1x=e^0=1x=e0=1.

Flashcard 15: Find the intersection xxx-value: f(x)=log⁡(x)f(x)=\log(x)f(x)=log(x) and g(x)=2g(x)=2g(x)=2.

Answer: x=100x=100x=100. Set log⁡(x)=2\log(x)=2log(x)=2: x=102=100x=10^2=100x=102=100.

Flashcard 16: Find the intersection xxx-value: f(x)=log⁡2(x)f(x)=\log_2(x)f(x)=log2​(x) and g(x)=5g(x)=5g(x)=5.

Answer: x=32x=32x=32. Set log⁡2(x)=5\log_2(x)=5log2​(x)=5: x=25=32x=2^5=32x=25=32.

Flashcard 17: Find the intersection xxx-value: f(x)=ln⁡(x)f(x)=\ln(x)f(x)=ln(x) and g(x)=1g(x)=1g(x)=1.

Answer: x=ex=ex=e. Set ln⁡(x)=1\ln(x)=1ln(x)=1: x=e1=ex=e^1=ex=e1=e.

Flashcard 18: What is the typical graphical output used to approximate solutions to f(x)=g(x)f(x)=g(x)f(x)=g(x)?

Answer: The intersection point(s) of the two graphs. Graph intersections visually show where functions are equal.

Flashcard 19: Identify the intersection xxx-value: f(x)=2x+1f(x)=2x+1f(x)=2x+1 and g(x)=7g(x)=7g(x)=7.

Answer: x=3x=3x=3. Set 2x+1=72x+1=72x+1=7 and solve: 2x=62x=62x=6, so x=3x=3x=3.

Flashcard 20: Identify the intersection xxx-value: f(x)=3x−2f(x)=3x-2f(x)=3x−2 and g(x)=x+6g(x)=x+6g(x)=x+6.

Answer: x=4x=4x=4. Set 3x−2=x+63x-2=x+63x−2=x+6 and solve: 2x=82x=82x=8, so x=4x=4x=4.

Flashcard 21: Identify the intersection xxx-value: f(x)=−x+5f(x)=-x+5f(x)=−x+5 and g(x)=2x−1g(x)=2x-1g(x)=2x−1.

Answer: x=2x=2x=2. Set −x+5=2x−1-x+5=2x-1−x+5=2x−1 and solve: 6=3x6=3x6=3x, so x=2x=2x=2.

Flashcard 22: What are the intersection xxx-values of f(x)=x2f(x)=x^2f(x)=x2 and g(x)=4g(x)=4g(x)=4?

Answer: x=−2x=-2x=−2 and x=2x=2x=2. Set x2=4x^2=4x2=4 and solve: x=±4=±2x=\pm\sqrt{4}=\pm 2x=±4​=±2.

Flashcard 23: What are the intersection xxx-values of f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=0g(x)=0g(x)=0?

Answer: x=−1x=-1x=−1 and x=1x=1x=1. Set x2−1=0x^2-1=0x2−1=0 and factor: (x−1)(x+1)=0(x-1)(x+1)=0(x−1)(x+1)=0.

Flashcard 24: What are the intersection xxx-values of f(x)=x2f(x)=x^2f(x)=x2 and g(x)=xg(x)=xg(x)=x?

Answer: x=0x=0x=0 and x=1x=1x=1. Set x2=xx^2=xx2=x and solve: x2−x=0x^2-x=0x2−x=0, so x(x−1)=0x(x-1)=0x(x−1)=0.

Flashcard 25: Find the intersection xxx-values: f(x)=x2+2xf(x)=x^2+2xf(x)=x2+2x and g(x)=0g(x)=0g(x)=0.

Answer: x=−2x=-2x=−2 and x=0x=0x=0. Set x2+2x=0x^2+2x=0x2+2x=0 and factor: x(x+2)=0x(x+2)=0x(x+2)=0.

Flashcard 26: Find the intersection xxx-values: f(x)=x2−4xf(x)=x^2-4xf(x)=x2−4x and g(x)=0g(x)=0g(x)=0.

Answer: x=0x=0x=0 and x=4x=4x=4. Set x2−4x=0x^2-4x=0x2−4x=0 and factor: x(x−4)=0x(x-4)=0x(x−4)=0.

Flashcard 27: Find the intersection xxx-values: f(x)=∣x∣f(x)=|x|f(x)=∣x∣ and g(x)=2g(x)=2g(x)=2.

Answer: x=−2x=-2x=−2 and x=2x=2x=2. Set ∣x∣=2|x|=2∣x∣=2: x=2x=2x=2 or x=−2x=-2x=−2.

Flashcard 28: Find the intersection xxx-values: f(x)=∣x−1∣f(x)=|x-1|f(x)=∣x−1∣ and g(x)=3g(x)=3g(x)=3.

Answer: x=−2x=-2x=−2 and x=4x=4x=4. Set ∣x−1∣=3|x-1|=3∣x−1∣=3: x−1=3x-1=3x−1=3 or x−1=−3x-1=-3x−1=−3.

Flashcard 29: Find the intersection xxx-value(s): f(x)=∣x∣f(x)=|x|f(x)=∣x∣ and g(x)=xg(x)=xg(x)=x.

Answer: x≥0x\ge 0x≥0. When x≥0x\geq 0x≥0, ∣x∣=x|x|=x∣x∣=x, so they're equal.

Flashcard 30: Find the intersection xxx-value(s): f(x)=∣x∣f(x)=|x|f(x)=∣x∣ and g(x)=−xg(x)=-xg(x)=−x.

Answer: x≤0x\le 0x≤0. When x≤0x\leq 0x≤0, ∣x∣=−x|x|=-x∣x∣=−x, so they're equal.