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

Verified step by step guidance

1. Start by writing the function as an equation with y: \(y = 2 \ln(3x)\).
2. To find the inverse, swap x and y: \(x = 2 \ln(3y)\).
3. Isolate the logarithm by dividing both sides by 2: \(\frac{x}{2} = \ln(3y)\).
4. Rewrite the logarithmic equation in exponential form: \(e^{\frac{x}{2}} = 3y\).
5. 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.

Key Concepts

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.