Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^x-5

Verified step by step guidance

1. Start by writing the function as an equation with y: \(y = e^{x-5}\).
2. To find the inverse, swap x and y: \(x = e^{y-5}\).
3. Solve for y by taking the natural logarithm (ln) of both sides: \(\ln(x) = y - 5\).
4. Isolate y to express the inverse function: \(y = \ln(x) + 5\), so \(f^{-1}(x) = \ln(x) + 5\).
5. Determine the domain and range: since \(f(x) = e^{x-5}\) has domain \((-\infty, \infty)\) and range \((0, \infty)\), the inverse \(f^{-1}(x)\) has domain \((0, \infty)\) and range \((-\infty, \infty)\).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. To find ƒ⁻¹(x), you solve the equation y = ƒ(x) for x in terms of y, then interchange x and y. This process helps determine the function that 'undoes' ƒ(x).

Exponential Functions and Their Properties

Exponential functions have the form f(x) = a^(x), where the variable is in the exponent. They are one-to-one and always positive, which ensures the existence of an inverse. Understanding their behavior, such as growth and domain restrictions, is essential for finding inverses.

Domain and Range of Functions and Their Inverses

The domain is the set of all possible inputs, and the range is the set of all possible outputs of a function. For inverse functions, the domain and range swap roles compared to the original function. Identifying these sets is crucial for correctly defining ƒ⁻¹(x).