Final answer:
A function must be one-to-one to have an inverse, meaning each output is associated with exactly one input. Inverse functions 'undo' the original function, such as subtraction being the inverse of addition.
Step-by-step explanation:
To have an inverse function, f must be one-to-one, so f(a) = f(b) implies a = b. In other words, each value of the function's output corresponds to exactly one input. This characteristic is essential for the existence of an inverse since the inverse function reverses the roles of inputs and outputs, ensuring the function can 'undo' the effect of the original function reliably. Take, for instance, the square function (f(x) = x2) and its inverse, the square root (f-1(x) = √x).
Inverse functions are a fundamental concept in algebra and calculus, where functions like the natural logarithm and the exponential function are inverses of each other. This can be seen with ex and ln(x), where the application of ln to ex will yield the original variable x. Similarly, applying the exponential function to ln(x) 'undoes' the logarithm and returns x.
The concept of inverses also applies to basic arithmetic operations. For example, subtraction is the inverse of addition, and division is the inverse of multiplication. These fundamental inverse relationships enable us to solve equations and manipulate expressions in algebra.