What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?

Verified step by step guidance

1. Recall that the line \(y = x\) has a slope of 1. Lines perpendicular to \(y = x\) have slopes that are the negative reciprocal of 1, which is \(-1\).
2. Consider a function \(f(x)\) whose graph is a line perpendicular to \(y = x\). Such a function can be written as \(f(x) = -x + b\), where \(b\) is the y-intercept.
3. To find the inverse function \(f^{-1}(x)\), start by replacing \(f(x)\) with \(y\): \(y = -x + b\). Then, swap \(x\) and \(y\) to get \(x = -y + b\).
4. Solve the equation \(x = -y + b\) for \(y\) to find the inverse function: \(y = -x + b\). Notice that the inverse has the same form as the original function.
5. Conclude that the inverse of a function whose graph is a line perpendicular to \(y = x\) is another line with the same slope \(-1\), meaning the inverse function is also perpendicular to \(y = x\) and has the same slope as the original function.

Key Concepts

Inverse Functions and Their Graphs

The inverse of a function reverses the roles of inputs and outputs, swapping x and y. Graphically, the inverse function's graph is the reflection of the original function's graph across the line y = x.

Linearity and Slope of Functions

A linear function has the form y = mx + b, where m is the slope. The slope determines the angle of the line, and perpendicular lines have slopes that are negative reciprocals of each other, meaning if one slope is m, the other is -1/m.

Relationship Between Slopes of a Function and Its Inverse

The slope of the inverse function is the reciprocal of the original function's slope. Since the inverse reflects over y = x, if the original line is perpendicular to y = x (which has slope 1), its slope is -1, and the inverse will have slope -1 as well.