Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)^x

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Start with the given function: \(f(x) = \left(\frac{1}{3}\right)^x\).

To find the inverse function, first replace \(f(x)\) with \(y\): \(y = \left(\frac{1}{3}\right)^x\).

Swap the variables \(x\) and \(y\) to get the inverse relation: \(x = \left(\frac{1}{3}\right)^y\).

To solve for \(y\), take the logarithm of both sides. You can use the natural logarithm or logarithm base 10: \(\log\left(x\right) = \log\left(\left(\frac{1}{3}\right)^y\right)\).

Use the logarithm power rule to bring down the exponent: \(\log(x) = y \cdot \log\left(\frac{1}{3}\right)\). Then solve for \(y\): \(y = \frac{\log(x)}{\log\left(\frac{1}{3}\right)}\).

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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 means no two different inputs produce the same output, which is essential for the inverse function to exist.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, effectively 'undoing' the original function. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Exponential and Logarithmic Functions

Exponential functions like f(x) = (1/3)^x have inverses that are logarithmic functions. To find the inverse, you rewrite the equation in logarithmic form, using the base of the exponential to solve for x.

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)x