Finding inverses Find the inverse function.


ƒ(x) = 3x² + 1, for x ≤ 0

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). When finding an inverse function, it is crucial to consider the domain of the original function, as it affects the range of the inverse function. In this case, the restriction x ≤ 0 is important for determining the inverse.

Solving for x

To find the inverse of a function, one typically starts by replacing f(x) with y, then solving for x in terms of y. This often involves algebraic manipulation, such as isolating x on one side of the equation. Once x is expressed in terms of y, the inverse function can be written as f⁻¹(y) = x, and then it can be rewritten in terms of x.