Problem 90 - Algebra and Trigonometry

Question

Determining When a Linear Function Has an Inverse For the linear function \(f(x)=m x+b\) to be one-to-one, what must be true about its slope? If it is one-to-one, find its inverse. Is the inverse linear? If so, what is its slope?

Step-by-Step Solution

Answer

For \(f(x) = mx + b\) to be one-to-one, \(m \neq 0\). The inverse is linear with slope \(\frac{1}{m}\).

Step 1: Determine Condition for a One-to-One Function

A function is one-to-one if each output is produced by exactly one input. For linear functions of the form \(f(x) = mx + b\), if the slope \(m eq 0\), the function is one-to-one. This is because a non-zero slope ensures that the function continues to increase or decrease, never taking the same value twice.

Step 2: Ensure the Function is One-to-One

For \(f(x) = mx + b\) to be one-to-one, our function must have \(m eq 0\). This means it must have a non-zero slope.

Step 3: Find the Inverse Function

If \(f(x) = mx + b\) is one-to-one, we can find its inverse. Start by swapping \(x\) and \(y\) in the equation \(y = mx + b\):\[ x = my + b \]Solve for \(y\):\[ x - b = my \]\[ y = \frac{x - b}{m} \]So, the inverse function is \(f^{-1}(x) = \frac{x - b}{m}\).

Step 4: Determine if the Inverse is Linear and Its Slope

The inverse function \(f^{-1}(x) = \frac{x - b}{m}\) can be rewritten as \(f^{-1}(x) = \frac{1}{m}x - \frac{b}{m}\). This is in the form \(y = mx + c\), which shows that it is linear. The slope of the inverse function is \(\frac{1}{m}\).

Key Concepts

Linear Functions

A linear function is a mathematical function that produces a straight line when graphed. It can be represented by the equation \(f(x) = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. The slope \(m\) represents the rate of change or how steep the line is. Linear functions are fundamental in math because they model relationships where one quantity depends on another. They are widely used because they are easy to understand and can describe many real-world scenarios, such as calculating cost or predicting growth. Understanding linear functions lays the groundwork for exploring more complex functions in algebra and calculus.

One-to-One Functions

A one-to-one function is a special type of function where each input is associated with a unique output, meaning no two inputs will have the same output. This property is crucial because it allows a function to have an inverse that is also a function. For a linear function \(f(x) = mx + b\), being one-to-one means its slope \(m\) should not be zero.

• If \(m\) is positive, the function increases continuously, ensuring each output is unique.
• If \(m\) is negative, the function decreases continuously, again ensuring uniqueness.

The concept of one-to-one functions is important as it paves the way to function inversion, enabling us to find inverse functions for such linear relationships.

Slope of a Line

The slope of a line in a linear function, represented by \(m\) in \(f(x) = mx + b\), is a measure of how steep the line is. It shows the relationship between the change in \(y\) over the change in \(x\) (often called rise over run).

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope results in a horizontal line, indicating no change in \(y\) and thus not one-to-one.

The slope is central to the unique behavior of linear functions and their inverses because it impacts whether a function is one-to-one and hence invertible.

Function Inversion

Function inversion is the process of finding a function that 'undoes' the effect of another. For a function to have an inverse, it must be one-to-one.To find the inverse of a linear function \(f(x) = mx + b\), follow these steps:

• Start with the equation \(y = mx + b\).
• Swap \(x\) and \(y\) to get \(x = my + b\).
• Solve for \(y\) by isolating it on one side: \(y = \frac{x-b}{m}\).

This turns the function into \(f^{-1}(x) = \frac{x-b}{m}\), and the new slope will be \(\frac{1}{m}\), still staying linear. Function inversion helps in solving equations where we need to 'reverse' a function to find an unknown variable, and it also offers insight into how opposite values relate.