Problem 59 - Elementary and Intermediate Algebra

Question

Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. \(f(x)=2 x\)

Step-by-Step Solution

Answer

The inverse of \(f(x) = 2x\) is \(f^{-1}(x) = \frac{x}{2}\).

Step 1: Set up the equation for the inverse

To find the inverse of the function, we start by replacing the function notation \(f(x)\) with \(y\). So, we have \(y = 2x\).

Step 2: Swap \(x\) and \(y\)

To find the inverse, swap \(x\) and \(y\) in the equation: \(x = 2y\).

Step 3: Solve for \(y\)

Rearrange the equation to solve for \(y\): Divide both sides by 2 to isolate \(y\), getting \(y = \frac{x}{2}\).

Step 4: Write the inverse function

The inverse function is the expression we found for \(y\): \(f^{-1}(x) = \frac{x}{2}\).

Step 5: Graph the functions

Graph both \(f(x) = 2x\) and its inverse \(f^{-1}(x) = \frac{x}{2}\) on the same coordinate system. The original function is a line with slope 2, passing through the origin, while the inverse is a line with slope \(\frac{1}{2}\), also passing through the origin.

Step 6: Show the line of symmetry

Draw the line \(y = x\) on the graph, which acts as the line of symmetry for the function and its inverse. Both graphs will mirror each other across this line.

Key Concepts

Graphing Functions

Graphing functions is a way to visually understand the behavior of equations. For the function \( f(x) = 2x \), it is a simple linear equation. When graphing linear functions:

• Identify the slope, which is the coefficient of \( x \). In this case, it is \( 2 \).
• Understand that the y-intercept is \( 0 \), because there is no constant added to \( 2x \).
• Plot the line by starting from the origin \((0,0)\) and using the slope to mark points. For a slope of \( 2 \), move up two units for every one unit you move right along the x-axis.

Once the function \( f(x) = 2x \) is plotted, graph the inverse \( f^{-1}(x) = \frac{x}{2} \). This inverse function is also linear, with a slope of \( \frac{1}{2} \). It will also pass through the origin. Remember, these lines will meet each other at the point where their values are equal.

Line of Symmetry

The line of symmetry in graphs of functions and their inverses is very important. In the case of \( f(x) = 2x \) and its inverse \( f^{-1}(x) = \frac{x}{2} \), the line of symmetry is \( y = x \).Symmetry lines help us visualize how the inverse function is a reflection of the original function. You draw the line \( y = x \), which creates a clear reflection point for \( f(x) \) and \( f^{-1}(x) \). This line runs diagonally through the graph,

• It passes through the points where \( x = y \).
• Both the function and its inverse will mirror around this line.

Understanding this line of symmetry helps us know that if a point \((a, b)\) is on the function \( f(x) \), then \((b, a)\) will be on its inverse \( f^{-1}(x) \). This visual relationship emphasizes how these equations relate to each other.

Coordinate System

A coordinate system is essential for plotting functions and their inverses. It allows us to visualize both functions on the same graph. The basic components of the coordinate system include:

• The x-axis, a horizontal line where values increase to the right and decrease to the left.
• The y-axis, a vertical line where values increase upwards and decrease downwards.
• These axes intersect at the origin, the point \((0,0)\).

When using a coordinate system to plot \( f(x) = 2x \) and \( f^{-1}(x) = \frac{x}{2} \), start at the origin, as both functions pass through this point.Both functions and their inverse can be plotted using the coordinates dictated by their equations. This setup makes it easy to predict and check how each function behaves and where it lies relative to the line of symmetry. Using this system efficiently helps students grasp the relationship between a function and its inverse, providing a practical approach to understanding their graphs.