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AP Precalculus Flashcards: Inverses Of Exponential Functions

Study Inverses Of Exponential 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

Inverses Of Exponential Functions

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QUESTION
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Identify the inverse of f(x)=73x−1f(x) = 7^{3x-1}.

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ANSWER

The inverse is f−1(x)=13(log7(x)+1)f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1). Factor out coefficient from both terms in exponent.

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What this deck covers

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

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Flashcard 1: Identify the inverse of f(x)=73x−1f(x) = 7^{3x-1}.

Answer: The inverse is f−1(x)=13(log7(x)+1)f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1). Factor out coefficient from both terms in exponent.

Flashcard 2: Identify the inverse of f(x)=axf(x) = a^x.

Answer: The inverse is f−1(x)=loga(x)f^{-1}(x) = \text{log}_a(x). Base-aa logarithm undoes base-aa exponentiation.

Flashcard 3: What operation is the inverse of exponentiation?

Answer: The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.

Flashcard 4: Identify the inverse of f(x)=axf(x) = a^x.

Answer: The inverse is f−1(x)=loga(x)f^{-1}(x) = \text{log}_a(x). Base-aa logarithm undoes base-aa exponentiation.

Flashcard 5: Identify the inverse of f(x)=9x−4f(x) = 9^{x-4}.

Answer: The inverse is f−1(x)=log9(x)+4f^{-1}(x) = \text{log}_9(x) + 4. Subtract 4 from exponent, so add 4 to inverse.

Flashcard 6: Find the inverse of f(x)=ex+2f(x) = e^{x+2}.

Answer: The inverse is f−1(x)=ln(x)−2f^{-1}(x) = \text{ln}(x) - 2. Addition in exponent becomes subtraction in inverse.

Flashcard 7: Find the inverse of f(x)=122x+1f(x) = 12^{2x+1}.

Answer: The inverse is f−1(x)=12log12(x)−12f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}. Factor out coefficient from exponent terms.

Flashcard 8: Identify the inverse of f(x)=b2x+1f(x) = b^{2x+1}.

Answer: The inverse is f−1(x)=12logb(x)−12f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}. Factor out coefficient and shift from exponent.

Flashcard 9: What is the inverse of f(x)=103xf(x) = 10^{3x}?

Answer: The inverse is f−1(x)=13log10(x)f^{-1}(x) = \frac{1}{3} \text{log}_{10}(x). Coefficient 3 in exponent becomes divisor 13\frac{1}{3}.

Flashcard 10: What is the inverse function of f(x)=e2xf(x) = e^{2x}?

Answer: The inverse is f−1(x)=12ln(x)f^{-1}(x) = \frac{1}{2} \text{ln}(x). Coefficient in exponent becomes divisor in inverse.

Flashcard 11: Which function is the inverse of f(x)=4x−3f(x) = 4^{x-3}?

Answer: The inverse is f−1(x)=log4(x)+3f^{-1}(x) = \text{log}_4(x) + 3. Subtract 3 from exponent, so add 3 to inverse.

Flashcard 12: What is the inverse of the natural exponential function?

Answer: The inverse is the natural logarithm function. Natural log ln⁡(x)\ln(x) undoes exe^x.

Flashcard 13: Identify the inverse of f(x)=b2x+1f(x) = b^{2x+1}.

Answer: The inverse is f−1(x)=12logb(x)−12f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}. Factor out coefficient and shift from exponent.

Flashcard 14: Determine the inverse of f(x)=2x+3f(x) = 2^{x+3}.

Answer: The inverse is f−1(x)=log2(x)−3f^{-1}(x) = \text{log}_2(x) - 3. Addition in exponent becomes subtraction in inverse.

Flashcard 15: Identify the steps to find inverse of f(x)=axf(x) = a^x.

Answer: Switch xx and yy, solve for yy. Result: y=loga(x)y = \text{log}_a(x). Standard method for finding inverse functions.

Flashcard 16: Identify the inverse of f(x)=2xf(x) = 2^x.

Answer: The inverse is f−1(x)=log2(x)f^{-1}(x) = \text{log}_2(x). Base-2 logarithm undoes base-2 exponentiation.

Flashcard 17: Identify the inverse of f(x)=15x−2f(x) = 15^{x-2}.

Answer: The inverse is f−1(x)=log15(x)+2f^{-1}(x) = \text{log}_{15}(x) + 2. Subtract 2 from exponent, so add 2 to inverse.

Flashcard 18: Find the inverse of f(x)=7x−1f(x) = 7^{x-1}.

Answer: The inverse is f−1(x)=log7(x)+1f^{-1}(x) = \text{log}_7(x) + 1. Subtract 1 from exponent, so add 1 to inverse.

Flashcard 19: Convert y=2x−5y = 2^{x-5} to its inverse function.

Answer: The inverse is y=log2(x)+5y = \text{log}_2(x) + 5. Subtract 5 from exponent, so add 5 to inverse.

Flashcard 20: Find the inverse of f(x)=7x−1f(x) = 7^{x-1}.

Answer: The inverse is f−1(x)=log7(x)+1f^{-1}(x) = \text{log}_7(x) + 1. Subtract 1 from exponent, so add 1 to inverse.

Flashcard 21: What is the inverse of the natural exponential function?

Answer: The inverse is the natural logarithm function. Natural log ln⁡(x)\ln(x) undoes exe^x.

Flashcard 22: What is the inverse of f(x)=103xf(x) = 10^{3x}?

Answer: The inverse is f−1(x)=13log10(x)f^{-1}(x) = \frac{1}{3} \text{log}_{10}(x). Coefficient 3 in exponent becomes divisor 13\frac{1}{3}.

Flashcard 23: Find the inverse of f(x)=6x+1f(x) = 6^{x+1}.

Answer: The inverse is f−1(x)=log6(x)−1f^{-1}(x) = \text{log}_6(x) - 1. Addition in exponent becomes subtraction in inverse.

Flashcard 24: What is the inverse of the function f(x)=exf(x) = e^x?

Answer: The inverse is f−1(x)=ln(x)f^{-1}(x) = \text{ln}(x). Natural log undoes the natural exponential function.

Flashcard 25: Find the inverse of f(x)=5xf(x) = 5^x.

Answer: The inverse is f−1(x)=log5(x)f^{-1}(x) = \text{log}_5(x). Base-5 logarithm undoes base-5 exponentiation.

Flashcard 26: Which is the inverse of f(x)=9x−2f(x) = 9^{x-2}?

Answer: The inverse is f−1(x)=log9(x)+2f^{-1}(x) = \text{log}_9(x) + 2. Subtract 2 from exponent, so add 2 to inverse.

Flashcard 27: Convert y=2x−5y = 2^{x-5} to its inverse function.

Answer: The inverse is y=log2(x)+5y = \text{log}_2(x) + 5. Subtract 5 from exponent, so add 5 to inverse.

Flashcard 28: State the property used to find inverses of exponential functions.

Answer: Exponential and logarithmic functions are inverses. Each function undoes the other's operation.

Flashcard 29: What is the inverse of the function f(x)=exf(x) = e^x?

Answer: The inverse is f−1(x)=ln(x)f^{-1}(x) = \text{ln}(x). Natural log undoes the natural exponential function.

Flashcard 30: Find the inverse of f(x)=5xf(x) = 5^x.

Answer: The inverse is f−1(x)=log5(x)f^{-1}(x) = \text{log}_5(x). Base-5 logarithm undoes base-5 exponentiation.

Flashcard 31: What is the inverse of f(x)=23xf(x) = 2^{3x}?

Answer: The inverse is f−1(x)=13log2(x)f^{-1}(x) = \frac{1}{3} \text{log}_2(x). Coefficient 3 in exponent becomes divisor 13\frac{1}{3}.

Flashcard 32: State the property used to find inverses of exponential functions.

Answer: Exponential and logarithmic functions are inverses. Each function undoes the other's operation.

Flashcard 33: Convert y=ex−1y = e^{x-1} to its inverse function.

Answer: The inverse is y=ln(x)+1y = \text{ln}(x) + 1. Subtract 1 from exponent, so add 1 to inverse.

Flashcard 34: Identify the inverse of f(x)=9x−4f(x) = 9^{x-4}.

Answer: The inverse is f−1(x)=log9(x)+4f^{-1}(x) = \text{log}_9(x) + 4. Subtract 4 from exponent, so add 4 to inverse.

Flashcard 35: What is the inverse function of f(x)=e2xf(x) = e^{2x}?

Answer: The inverse is f−1(x)=12ln(x)f^{-1}(x) = \frac{1}{2} \text{ln}(x). Coefficient in exponent becomes divisor in inverse.

Flashcard 36: What operation is the inverse of exponentiation?

Answer: The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.

Flashcard 37: Find the inverse of f(x)=ex+2f(x) = e^{x+2}.

Answer: The inverse is f−1(x)=ln(x)−2f^{-1}(x) = \text{ln}(x) - 2. Addition in exponent becomes subtraction in inverse.

Flashcard 38: Identify the inverse of f(x)=15x−2f(x) = 15^{x-2}.

Answer: The inverse is f−1(x)=log15(x)+2f^{-1}(x) = \text{log}_{15}(x) + 2. Subtract 2 from exponent, so add 2 to inverse.

Flashcard 39: What is the inverse of f(x)=82xf(x) = 8^{2x}?

Answer: The inverse is f−1(x)=12log8(x)f^{-1}(x) = \frac{1}{2} \text{log}_8(x). Coefficient 2 in exponent becomes divisor 12\frac{1}{2}.

Flashcard 40: What is the inverse of f(x)=bxf(x) = b^x?

Answer: The inverse is f−1(x)=logb(x)f^{-1}(x) = \text{log}_b(x). General base-bb logarithm undoes base-bb exponentiation.

Flashcard 41: Convert y=5x−4y = 5^{x-4} to its inverse function.

Answer: The inverse is y=log5(x)+4y = \text{log}_5(x) + 4. Subtract 4 from exponent, so add 4 to inverse.

Flashcard 42: Identify the inverse of f(x)=73x−1f(x) = 7^{3x-1}.

Answer: The inverse is f−1(x)=13(log7(x)+1)f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1). Factor out coefficient from both terms in exponent.

Flashcard 43: What is the inverse of f(x)=13xf(x) = 13^x?

Answer: The inverse is f−1(x)=log13(x)f^{-1}(x) = \text{log}_{13}(x). Base-13 logarithm undoes base-13 exponentiation.

Flashcard 44: Find the inverse of f(x)=32xf(x) = 3^{2x}.

Answer: The inverse is f−1(x)=12log3(x)f^{-1}(x) = \frac{1}{2} \text{log}_3(x). Coefficient 2 in exponent becomes divisor 12\frac{1}{2}.

Flashcard 45: What is the inverse of f(x)=13xf(x) = 13^x?

Answer: The inverse is f−1(x)=log13(x)f^{-1}(x) = \text{log}_{13}(x). Base-13 logarithm undoes base-13 exponentiation.

Flashcard 46: Convert y=5x−4y = 5^{x-4} to its inverse function.

Answer: The inverse is y=log5(x)+4y = \text{log}_5(x) + 4. Subtract 4 from exponent, so add 4 to inverse.

Flashcard 47: Which function is the inverse of f(x)=3xf(x) = 3^x?

Answer: The inverse is f−1(x)=log3(x)f^{-1}(x) = \text{log}_3(x). Base-3 logarithm undoes base-3 exponentiation.

Flashcard 48: Find the inverse of f(x)=122x+1f(x) = 12^{2x+1}.

Answer: The inverse is f−1(x)=12log12(x)−12f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}. Factor out coefficient from exponent terms.

Flashcard 49: Convert y=4xy = 4^x to its inverse function.

Answer: The inverse is x=4yx = 4^y or y=log4(x)y = \text{log}_4(x). Switch variables and solve for yy.

Flashcard 50: What is the inverse of f(x)=82xf(x) = 8^{2x}?

Answer: The inverse is f−1(x)=12log8(x)f^{-1}(x) = \frac{1}{2} \text{log}_8(x). Coefficient 2 in exponent becomes divisor 12\frac{1}{2}.

Flashcard 51: What is the inverse of f(x)=bxf(x) = b^x?

Answer: The inverse is f−1(x)=logb(x)f^{-1}(x) = \text{log}_b(x). General base-bb logarithm undoes base-bb exponentiation.

Flashcard 52: Determine the inverse of f(x)=2x+3f(x) = 2^{x+3}.

Answer: The inverse is f−1(x)=log2(x)−3f^{-1}(x) = \text{log}_2(x) - 3. Addition in exponent becomes subtraction in inverse.

Flashcard 53: Find the inverse of f(x)=3x+5f(x) = 3^{x+5}.

Answer: The inverse is f−1(x)=log3(x)−5f^{-1}(x) = \text{log}_3(x) - 5. Addition in exponent becomes subtraction in inverse.

Flashcard 54: Find the inverse of f(x)=11x+2f(x) = 11^{x+2}.

Answer: The inverse is f−1(x)=log11(x)−2f^{-1}(x) = \text{log}_{11}(x) - 2. Addition in exponent becomes subtraction in inverse.

Flashcard 55: Find the inverse of f(x)=3x+5f(x) = 3^{x+5}.

Answer: The inverse is f−1(x)=log3(x)−5f^{-1}(x) = \text{log}_3(x) - 5. Addition in exponent becomes subtraction in inverse.

Flashcard 56: Which function is the inverse of f(x)=4x−3f(x) = 4^{x-3}?

Answer: The inverse is f−1(x)=log4(x)+3f^{-1}(x) = \text{log}_4(x) + 3. Subtract 3 from exponent, so add 3 to inverse.

Flashcard 57: Find the inverse of f(x)=6x+1f(x) = 6^{x+1}.

Answer: The inverse is f−1(x)=log6(x)−1f^{-1}(x) = \text{log}_6(x) - 1. Addition in exponent becomes subtraction in inverse.

Flashcard 58: Convert y=4xy = 4^x to its inverse function.

Answer: The inverse is x=4yx = 4^y or y=log4(x)y = \text{log}_4(x). Switch variables and solve for yy.

Flashcard 59: Identify the inverse of f(x)=2xf(x) = 2^x.

Answer: The inverse is f−1(x)=log2(x)f^{-1}(x) = \text{log}_2(x). Base-2 logarithm undoes base-2 exponentiation.

Flashcard 60: Identify the steps to find inverse of f(x)=axf(x) = a^x.

Answer: Switch xx and yy, solve for yy. Result: y=loga(x)y = \text{log}_a(x). Standard method for finding inverse functions.

Flashcard 61: Find the inverse of f(x)=32xf(x) = 3^{2x}.

Answer: The inverse is f−1(x)=12log3(x)f^{-1}(x) = \frac{1}{2} \text{log}_3(x). Coefficient 2 in exponent becomes divisor 12\frac{1}{2}.

Flashcard 62: Find the inverse of f(x)=11x+2f(x) = 11^{x+2}.

Answer: The inverse is f−1(x)=log11(x)−2f^{-1}(x) = \text{log}_{11}(x) - 2. Addition in exponent becomes subtraction in inverse.

Flashcard 63: Which is the inverse of f(x)=9x−2f(x) = 9^{x-2}?

Answer: The inverse is f−1(x)=log9(x)+2f^{-1}(x) = \text{log}_9(x) + 2. Subtract 2 from exponent, so add 2 to inverse.

Flashcard 64: Which function is the inverse of f(x)=3xf(x) = 3^x?

Answer: The inverse is f−1(x)=log3(x)f^{-1}(x) = \text{log}_3(x). Base-3 logarithm undoes base-3 exponentiation.

Flashcard 65: Convert y=ex−1y = e^{x-1} to its inverse function.

Answer: The inverse is y=ln(x)+1y = \text{ln}(x) + 1. Subtract 1 from exponent, so add 1 to inverse.

Flashcard 66: What is the inverse of f(x)=23xf(x) = 2^{3x}?

Answer: The inverse is f−1(x)=13log2(x)f^{-1}(x) = \frac{1}{3} \text{log}_2(x). Coefficient 3 in exponent becomes divisor 13\frac{1}{3}.

Flashcard 67: What is the inverse of f(x)=10xf(x) = 10^x?

Answer: The inverse is f−1(x)=log10(x)f^{-1}(x) = \text{log}_{10}(x). Common logarithm undoes base-10 exponentiation.

Flashcard 68: What is the inverse of f(x)=10xf(x) = 10^x?

Answer: The inverse is f−1(x)=log10(x)f^{-1}(x) = \text{log}_{10}(x). Common logarithm undoes base-10 exponentiation.