Find the inverse of the function f(x)=mx, where m is a constant different from zero.

Verified step by step guidance

1. Start with the given function: \(f(x) = mx\), where \(m \neq 0\).
2. To find the inverse function, replace \(f(x)\) with \(y\): \(y = mx\).
3. Swap the roles of \(x\) and \(y\) to find the inverse: \(x = my\).
4. Solve this equation for \(y\) by dividing both sides by \(m\): \(y = \frac{x}{m}\).
5. Rewrite \(y\) as the inverse function notation: \(f^{-1}(x) = \frac{x}{m}\).

Key Concepts

Definition of an Inverse Function

An inverse function reverses the effect of the original function, mapping outputs back to their inputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, meaning applying one after the other returns the original value.

One-to-One Functions and Invertibility

A function must be one-to-one (injective) to have an inverse, ensuring each output corresponds to exactly one input. For f(x) = mx with m ≠ 0, the function is linear and strictly monotonic, guaranteeing it is invertible.

Finding the Inverse of a Linear Function

To find the inverse of f(x) = mx, solve the equation y = mx for x in terms of y. This involves isolating x, resulting in x = y/m, which defines the inverse function f⁻¹(x) = x/m.