Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 5^x + 1

검증된 단계별 안내

Start with the given function: \(f(x) = 5^x + 1\).

To find the inverse function, first replace \(f(x)\) with \(y\): \(y = 5^x + 1\).

Swap the variables \(x\) and \(y\) to reflect the inverse relationship: \(x = 5^y + 1\).

Solve this equation for \(y\) by isolating the exponential term: \(x - 1 = 5^y\).

Take the logarithm base 5 of both sides to solve for \(y\): \(y = \log_5(x - 1)\), which gives the inverse function \(f^{-1}(x) = \log_5(x - 1)\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

One-to-One Functions

A one-to-one function is a function where each output corresponds to exactly one input, ensuring it has an inverse. This property is essential because only one-to-one functions have inverses that are also functions. Verifying this helps confirm that the inverse function exists.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. To find the inverse, you replace f(x) with y, interchange x and y, then solve for y. The inverse function undoes the original function's operation.

Exponential and Logarithmic Functions

Since the given function involves an exponential expression (5^x), its inverse will involve a logarithm with base 5. Understanding that logarithms are the inverses of exponential functions is crucial for solving for the inverse function.

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 5x + 1