Problem 47 - Thomas Calculus in SI Units

Question

a. Find the inverse of \(f(x)=x+1 .\) Graph \(f\) and its inverse together. Add the line \(y=x\) to your sketch, drawing it with dashes or dots for contrast. b. Find the inverse of \(f(x)=x+b(b\) constant). How is the graph of \(f^{-1}\) related to the graph of \(f ?\) c. What can you conclude about the inverses of functions whose graphs are lines parallel to the line \(y=x ?\)

Step-by-Step Solution

Answer

The inverse of \(f(x) = x + 1\) is \(f^{-1}(x) = x - 1\). For \(f(x) = x + b\), the inverse is \(f^{-1}(x) = x - b\), both are symmetric about \(y = x\). Lines parallel to \(y=x\) remain symmetric with their inverses.

Step 1: Finding the Inverse of f(x) = x + 1

To find the inverse function, we need to solve the equation for \(x\). Start by replacing \(f(x)\) with \(y\): \(y = x + 1\). To find the inverse, switch \(x\) and \(y\): \(x = y + 1\). Then solve for \(y\): \(y = x - 1\). Thus, the inverse function is \(f^{-1}(x) = x - 1\).

Step 2: Graphing f(x) and its Inverse

Plot the function \(f(x) = x + 1\). This is a straight line with a slope of 1 and a y-intercept at (0, 1). Next, plot the inverse function \(f^{-1}(x) = x - 1\). This line also has a slope of 1, but with a y-intercept at (0, -1). The line \(y = x\) is a dashed line that serves as a reflection line between \(f(x)\) and \(f^{-1}(x)\).

Step 3: Finding the Inverse of f(x) = x + b

For the function \(f(x) = x + b\), follow a similar process: replace \(f(x)\) with \(y\): \(y = x + b\), then exchange \(x\) and \(y\): \(x = y + b\), and solve for \(y\): \(y = x - b\). Thus, \(f^{-1}(x) = x - b\).

Step 4: Describing the Relationship Between f and f^{-1}

The inverse function \(f^{-1}(x) = x - b\) is a reflection of the original function \(f(x) = x + b\) across the line \(y = x\). Since both functions are linear with the same slope (1), their graphs are symmetric about the line \(y = x\).

Step 5: Conclusion on Inverses for Lines Parallel to y = x

Lines parallel to \(y = x\) have the form \(f(x) = ax + c\) where \(a = 1\). These functions and their inverses (\(f^{-1}(x) = x - c\)), are symmetric across the line \(y = x\). As a result, if a line is parallel to \(y = x\), its inverse will maintain this symmetry and parallelism.

Key Concepts

Graphing Functions

Graphing functions is a visual way to understand how a function behaves. For linear functions like \( f(x) = x + 1 \), graphing means drawing a straight line on a coordinate plane. The equation tells us:

• The slope of the line, which here is 1. This means the line goes up one step for every step it goes right.
• The y-intercept, which is where the line crosses the y-axis. For \( f(x) = x + 1 \), the intercept is at \( y = 1 \).

To graph this function, start at the point (0, 1) on the y-axis. From there, use the slope to find other points on the line. Draw a line through these points to get the graph of \( f(x) = x + 1 \).
To add the inverse \( f^{-1}(x) = x - 1 \), repeat these steps but start at (0, -1). The slope remains 1, so this line will be parallel to the original function but shifted down.
Graphing the function and its inverse together demonstrates their relationship and symmetry around the line \( y = x \).

Reflection Symmetry

Reflection symmetry is a fascinating concept in geometry and function analysis. When we talk about an inverse function being a reflection of the original function, we're saying the graph 'flips' across a certain line. For linear functions like \( f(x) = x + b \) and its inverse \( f^{-1}(x) = x - b \), this line is \( y = x \).
Imagine the line \( y = x \) as a mirror: everything above the line reflects below, and vice versa. This symmetry is crucial because it visually confirms that the equations truly are inverses. If you fold your graph along \( y = x \), the two lines should match up perfectly.
This reflection symmetry makes it easier to understand why the expressions are inverses. By switching \( x \) and \( y \) and solving, you effectively 'mirror' the function, creating a direct and visual confirmation of the inverse function.

Linear Functions

Linear functions are equations of straight lines, and they follow a simple format: \( f(x) = ax + b \) where \( a \) and \( b \) are constants.

• \( a \) is the slope, dictating the line's steepness.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

For linear functions of the form \( f(x) = x + b \), the slope is 1, which means the line rises evenly with no tilt. To find an inverse, swap the positions of \( x \) and \( y \), then resolve for \( y \). For example, the inverse of \( f(x) = x + b \) is \( f^{-1}(x) = x - b \).
This change in sign for \( b \) indicates that the inverse function translates in the opposite direction of the original. When graphed, these lines not only appear parallel but are also symmetric except through different y-intercept points, maintaining the same slope. Recognizing these patterns is key in understanding linear functions and their inverses.