A. Find the inverse of f(x)=-x+1 . Graph the line y=-x+1...

a. Find the inverse of $f(x)=-x+1 .$ Graph the line $y=-x+1$ together with the line $y=x$ . At what angle do the lines intersect? b. Find the inverse of $f(x)=-x+b(b$ constant $) .$ What angle does the line $y=-x+b$ make with the line $y=x ?$ c. What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line $y=x ?$

a. Find the inverse of $f(x)=-x+1 .$ Graph the line $y=-x+1$ together with the line $y=x$ . At what angle do the lines intersect?
b. Find the inverse of $f(x)=-x+b(b$ constant $) .$ What angle does the line $y=-x+b$ make with the line $y=x ?$
c. What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line $y=x ?$

Key Concepts

Inverse Function

An inverse function reverses the roles of inputs and outputs; that is, if f takes x to y, then its inverse f?¹ takes y back to x. In practical terms, finding the inverse involves algebraically solving for x in terms of y and then interchanging the variables. This guarantees that f(f?¹(x)) = x for every x in the domain of f?¹ and f?¹(f(x)) = x for every x in the domain of f.

Reflection Across the Line y = x

The graphical connection between a function and its inverse is that the graph of the inverse is a reflection of the graph of the function across the line y = x. This means that if you were to fold the coordinate plane along the line y = x, the graph of f would coincide exactly with the graph of f?¹.

Linear Functions and Their Graphs

A linear function is one of the simplest forms of functions represented by a straight line, characterized by its slope and y-intercept. The general form y = mx + b encapsulates its properties, where m controls the steepness and direction of the line and b determines its vertical position. Understanding linear functions is essential in exploring the relationships between lines, such as parallelism, perpendicularity, and reflection symmetry.

Angle Between Lines

The angle at which two lines intersect can be determined from their slopes. Specifically, for lines with slopes m? and m?, the tangent of the angle of intersection ? is given by |(m? - m?) / (1 + m?m?)|. This relationship is particularly useful for finding the angle between any two lines, including identifying when they intersect perpendicularly (i.e., at 90°).

Perpendicularity and Self-Inverse Functions

When a line is perpendicular to the line y = x, its slope must be -1 because the line y = x has a slope of 1 and the product of the slopes of two perpendicular lines is -1. As a result, any linear function with a slope of -1 is symmetric with respect to the line y = x, making it a self-inverse function. This means that its own inverse is identical to the function itself, a fact that follows directly from its reflective symmetry.

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