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

Verified step by step guidance

1. Step 1: Start by setting $y=f(x)=2\ln (3x)$. We want to find the inverse function, so we need to solve for $x$ in terms of $y$.
2. Step 2: Divide both sides of the equation by 2 to isolate the natural logarithm: $\frac{y}{2}=\ln (3x)$.
3. Step 3: Exponentiate both sides to eliminate the natural logarithm: $e^{\frac{y}{2}}=3x$.
4. Step 4: Solve for $x$ by dividing both sides by 3: $x=\frac{e^{\frac{y}{2}}}{3}$.
5. Step 5: Replace $y$ with $x$ to express the inverse function: $f^{-1}(x)=\frac{e^{\frac{x}{2}}}{3}$. Determine the domain and range: The domain of $f(x)$ is $x>0$, and the range is all real numbers. Therefore, the domain of $f^{-1}(x)$ is all real numbers, and the range is $x>0$.

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function. For a function f(x), its inverse f^-1(x) satisfies the equation f(f^-1(x)) = x. To find the inverse, we typically solve for x in terms of y, then swap the variables. Understanding how to derive and interpret inverse functions is crucial for solving problems involving them.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.718. It is the inverse operation of exponentiation with base e. The function ln(x) is defined for x > 0, and it is important to recognize its properties, such as its domain and range, when working with logarithmic functions.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). For the function f(x) = 2 ln(3x), the domain is x > 0 due to the logarithm, and the range is all real numbers since the output can extend infinitely in both directions. Understanding these concepts is essential for analyzing functions and their inverses.