Let y=f(x)=m x+b m ≠0 (a) Writing to Learn Give a convincing...

Let $y=f(x)=m x+b$ $m \neq 0$ (a) Writing to Learn Give a convincing argument that $f$ is a one-to-one function. (b) Find a formula for the inverse of $f .$ How are the slopes of $f$ and $f^{-1}$ related? (c) If the graphs of two functions are parallel lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? (d) If the graphs of two functions are perpendicular lines with a nonzero slope, what can you say about the graphs of the inverses of the functions?

Let $y=f(x)=m x+b$ $m \neq 0$
(a) Writing to Learn Give a convincing argument that $f$ is a one-to-one function.
(b) Find a formula for the inverse of $f .$ How are the slopes of $f$ and $f^{-1}$ related?
(c) If the graphs of two functions are parallel lines with a nonzero slope, what can you say about the graphs of the inverses of the functions?
(d) If the graphs of two functions are perpendicular lines with a nonzero slope, what can you say about the graphs of the inverses of the functions?

Key Concepts

One-to-One Function

A one-to-one function is one in which every output is associated with exactly one input. This property, also known as injectivity, ensures that the function passes the horizontal line test, meaning no horizontal line intersects the graph more than once. In the context of linear functions with a nonzero slope, the strict monotonicity (either strictly increasing or strictly decreasing) guarantees that the function is one-to-one.

Inverse Function

An inverse function reverses the effect of the original function by swapping the roles of inputs and outputs. Finding an inverse typically involves replacing y with x and solving for y. For linear functions, this process yields another linear function, and the inverse function effectively undoes the action of the original function.

Slope Relationship of Linear Functions and Their Inverses

When dealing with linear functions in slope-intercept form, the slope of the inverse function is closely related to the slope of the original function. Specifically, if the original function has a slope m (with m not equal to zero), then the inverse function will have a slope of 1/m. This reciprocal relationship arises from the process of solving for the inverse and reflects the geometric reflection of the graph about the line y = x.

Parallel Lines

Two lines are parallel if they have the same slope. In the context of functions and their inverses, if the graphs of two functions (which are linear with nonzero slopes) are parallel, then because their slopes are equal, the inverses of these functions will also have equal slopes (since each inverse's slope is the reciprocal of the same number) and hence will be parallel as well.

Perpendicular Lines

Two lines are perpendicular if their slopes multiply to -1, meaning they are negative reciprocals of each other. For linear functions, if the originals are perpendicular, the relationship between their slopes will manifest in their inverses such that the slopes of the inverse functions will also be negative reciprocals of each other. This is because taking the reciprocal of each slope preserves the negative reciprocal relationship, thus ensuring that the graphs of the inverses are still perpendicular.

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