Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 5^x + 1

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output is produced by exactly one input. This means that no two different inputs can yield the same output. This property is crucial for finding the inverse of a function, as only one-to-one functions have inverses that are also functions.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function ƒ takes an input x and produces an output y, the inverse function ƒ⁻¹ takes y and returns x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y.

Exponential Functions

Exponential functions are mathematical functions of the form ƒ(x) = a^x, where a is a positive constant. In the given function ƒ(x) = 5^x + 1, the term 5^x represents the exponential part, and understanding its properties is essential for finding the inverse, which often involves logarithms.