Problem 36 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The graph of the inverse is a line with slope \( \frac{1}{m} \) and y-intercept \( -\frac{b}{m} \).

Step 1: Understand the Function

The given function is a linear function represented by the equation \( f(x) = mx + b \). The parameters \( m \) and \( b \) are constants with \( m eq 0 \).

Step 2: Express As an Equation

To find the inverse, start by considering \( y = f(x) = mx + b \). Next, exchange \( x \) and \( y \) to swap their roles: \( x = my + b \).

Step 3: Solve for y

Rearrange the equation \( x = my + b \) to solve for \( y \) in terms of \( x \). Begin by subtracting \( b \) from both sides to get \( x - b = my \), and then divide both sides by \( m \): \( y = \frac{x - b}{m} \). This is the inverse function \( f^{-1}(x) \).

Step 4: Identify the Slope and Y-Intercept

The inverse function is \( y = \frac{1}{m}x - \frac{b}{m} \). This equation is in the slope-intercept form \( y = mx + c \), where the slope is \( \frac{1}{m} \) and the y-intercept is \( -\frac{b}{m} \).

Step 5: Conclusion

The inverse function is a line with a slope of \( \frac{1}{m} \) and a y-intercept of \( -\frac{b}{m} \). This confirms that the graph of the inverse function is a line with the required properties.

Key Concepts

Understanding Linear Functions

Linear functions are equations of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. The constant \(m\) is known as the slope, and \(b\) is the y-intercept. These functions produce straight lines when graphed because the relationship between \(x\) and \(f(x)\) is direct and proportionate.
Here are the main points of linear functions:

• Linear functions describe a straight line in a coordinate plane.
• The graph of a linear function will always be a line unless the slope \(m = 0\), in which case it's a horizontal line.
• Each increase in \(x\) by 1 results in an increase or decrease in \(f(x)\) by \(m\).

Understanding these basic properties is vital, as they form the foundation for grasping more complex concepts like inverse functions.

Exploring Slope-Intercept Form

The slope-intercept form is a particular way of expressing linear equations. It is written as \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) is the y-intercept, which is the point where the line crosses the y-axis.
The slope \(m\) indicates how steep the line is. For every one unit increase in \(x\), \(y\) increases by \(m\). If \(m\) is positive, the line rises from left to right; if negative, it falls.
The y-intercept \(c\) tells us where the line crosses the y-axis. When \(x = 0\), \(y = c\). This helps us quickly locate the starting point of the line on a graph.

• The slope \(m = \frac{\text{rise}}{\text{run}}\). This is a ratio that describes how much \(y\) changes relative to changes in \(x\).
• The y-intercept \(c\) is particularly useful in graphing because it gives the initial value of \(y\) when \(x\) is zero.

This form is incredibly useful for quickly sketching the graph of a linear equation.

Finding Inverse Functions

Finding the inverse of a function involves reversing the roles of the input and output values. For a linear function \(f(x) = mx + b\), the process of finding its inverse involves finding \(x\) in terms of \(y\), then swapping them to solve for the new \(y\).
Here's how you can find the inverse:

• Begin by rewriting the function in terms of \(y\): \(y = mx + b\).
• Swap \(x\) and \(y\) to get the equation \(x = my + b\).
• Solve for \(y\) by first isolating \(my\): \(x - b = my\). Then divide each side by \(m\) to get \(y = \frac{x - b}{m}\).

Now you have the inverse function \(f^{-1}(x) = \frac{1}{m}x - \frac{b}{m}\), which is also in the slope-intercept form. The slope is now \(\frac{1}{m}\), and the y-intercept is \(-\frac{b}{m}\).
Inverse functions are useful for determining input values from given output values. They are essential for problem-solving and understanding how changes in input affect output in reverse.