Problem 43 - Intermediate Algebra

Question

(a) find the inverse of the given function, and (b) graph the given function and its inverse on the same set of axes. (Objective 4) $$f(x)=-\frac{1}{3} x$$

Step-by-Step Solution

Answer

The inverse of \( f(x) = -\frac{1}{3}x \) is \( f^{-1}(x) = -3x \). Graph both \( f(x) = -\frac{1}{3}x \) and \( f^{-1}(x) = -3x \) on the same axes.

Step 1: Understand the Given Function

The given function is \( f(x) = -\frac{1}{3}x \). This is a linear function with a slope of \(-\frac{1}{3}\) and a y-intercept of 0.

Step 2: Find the Inverse of the Function

To find the inverse, let's first rewrite the function as \( y = -\frac{1}{3}x \). To find the inverse, swap \( x \) and \( y \) to get \( x = -\frac{1}{3}y \). Now solve for \( y \) by multiplying both sides by \(-3\): \( y = -3x \). Thus, the inverse function is \( f^{-1}(x) = -3x \).

Step 3: Graph the Original Function

To graph \( f(x) = -\frac{1}{3}x \), note that it is a straight line with slope \(-\frac{1}{3}\). The line crosses the origin (0,0). To visualize, choose another point such as \( (3, -1) \), since when \( x=3 \), \( -\frac{1}{3} \times 3 = -1 \).

Step 4: Graph the Inverse Function

Graph \( f^{-1}(x) = -3x \), which is also a straight line. This function also passes through the origin and has a slope of \(-3\). Choose a second point like \( (1, -3) \), because \(-3 \times 1 = -3\).

Step 5: Confirm the Graph with Reflection

Check that the graphs of the function and its inverse are reflections across the line \( y = x \). Each point \( (a, b) \) on \( f(x) \) has a corresponding point \( (b, a) \) on \( f^{-1}(x) \). For example, from point \( (3, -1) \) on \( f(x) \), we expect \( (-1, 3) \) on \( f^{-1}(x) \).

Step 6: Final Graphing Details

Plot all points derived in the steps above correctly in their respective domains. Combine both functions on a single graph so comparisons and reflections of each point can visually validate the inverse.

Key Concepts

Graphing Linear Functions

When graphing linear functions, understanding the basic concepts of slope and intercept is essential. Linear functions can be represented in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises, while a negative slope falls.

In our exercise, the function \( f(x) = -\frac{1}{3}x \) has a slope of \(-\frac{1}{3}\). This indicates that for each increase of 3 units in the x-direction, the function decreases by 1 unit in the y-direction. The y-intercept is 0, which means the line crosses the origin. To graph such a function, start at the origin, move 3 units right, and then go down 1 unit for another point to draw the line.

• Start from the origin (0,0).
• Move horizontally 3 units to the right.
• Move vertically 1 unit down.
• Draw a line through these points.

This results in a straight line that depicts the function on a graph.

Linear Algebra

Linear algebra is a vast field of mathematics often employed to solve systems of linear equations. It deals with vectors, vector spaces, linear transformations, and matrices. When studying inverse functions and graphing, linear algebra concepts prove helpful as they provide tools for understanding transformations and reflections on a graph.

In the context of finding an inverse function, linear algebra simplifies the process of swapping variables. For the function \( f(x) = -\frac{1}{3}x \), finding the inverse requires the transformation where you swap x and y, yielding \( x = -\frac{1}{3}y \). Solving this equation for y involves multiplying both sides by -3, leading us to the inverse function \( f^{-1}(x) = -3x \).

• Understand the nature of linear transformations.
• Swap variables to find the inverse function.
• Rearrange to solve for the new output variable.

Linear algebra provides a structure to systematically approach these transformations, allowing a deeper understanding of function inverses.

Function Reflection

In relation to functions, reflection often refers to visualizing how one function relates to another, especially in the context of inverse functions. In mathematics, two functions are inverses if their graphs are reflections of each other across the line \( y = x \).

When reflecting the function \( f(x) = -\frac{1}{3}x \) over the line \( y = x \), we obtain its inverse \( f^{-1}(x) = -3x \). This reflection process is crucial because each point \( (a, b) \) on the graph of \( f(x) \) corresponds to a point \( (b, a) \) on \( f^{-1}(x) \).

• Identify any point on \( f(x) \).
• Swap coordinates for the related point on \( f^{-1}(x) \).
• Verify the reflection symmetry along line \( y = x \).

To confirm the reflection visually, plot both lines on the same axes. They should mirror each other across this line. This conceptual reflection underlines the nature of inverse functions graphically.