Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.

Verified step by step guidance

1. Start with the given function: $f(x) = mx + b$, where \(m\) and \(b\) are constants and \(m \neq 0\).
2. To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): $y = mx + b$.
3. Swap the roles of \(x\) and \(y\) to find the inverse: $x = my + b$.
4. Solve this equation for \(y\) to express the inverse function: subtract \(b\) from both sides to get $x - b = my$, then divide both sides by \(m\) to isolate \(y\): \(y = \frac{x - b}{m}\).
5. Rewrite the inverse function in slope-intercept form: \(f^{-1}(x) = \frac{1}{m}x - \frac{b}{m}\). This shows the inverse is a line with slope \(\frac{1}{m}\) and \(y\)-intercept \(-\frac{b}{m}\).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x. Understanding how to find and interpret inverses is essential for analyzing the relationship between a function and its inverse graphically.

Linear Functions and Their Properties

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. The graph is a straight line, and the slope indicates the rate of change. Recognizing how slope and intercept affect the line helps in determining the characteristics of its inverse.

Finding the Inverse of a Linear Function

To find the inverse of f(x) = mx + b, solve for x in terms of y, then interchange variables. This process yields f⁻¹(x) = (1/m)x - b/m, showing the inverse is also linear with slope 1/m and y-intercept -b/m. This algebraic manipulation is key to proving the inverse's graph properties.