Find ƒ^-1(x), and give the domain and range. ƒ(x) = 2 ln 3x

검증된 단계별 안내

Start by writing the function as an equation with y: \(y = 2 \ln(3x)\).

To find the inverse, swap x and y: \(x = 2 \ln(3y)\).

Isolate the logarithm by dividing both sides by 2: \(\frac{x}{2} = \ln(3y)\).

Rewrite the logarithmic equation in exponential form: \(e^{\frac{x}{2}} = 3y\).

Solve for y to get the inverse function: \(y = \frac{e^{\frac{x}{2}}}{3}\). Then, determine the domain and range by considering the original function's domain and range and how they switch for the inverse.

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.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. To find ƒ⁻¹(x), you replace ƒ(x) with y, interchange x and y, then solve for y. The inverse exists only if the original function is one-to-one.

Properties of Logarithmic Functions

Logarithmic functions, like ln(x), are the inverses of exponential functions. The natural logarithm ln(x) is defined only for x > 0, and it has a domain of (0, ∞) and range of (-∞, ∞). Understanding these properties helps determine the domain and range of the function and its inverse.

Domain and Range of Functions

The domain is the set of all possible input values, and the range is the set of all possible output values of a function. When finding an inverse, the domain of the original function becomes the range of the inverse, and vice versa. Identifying these sets ensures the function and its inverse are properly defined.

Find ƒ-1(x), and give the domain and range. ƒ(x) = 2 ln 3x