Problem 18 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ G(x)=|x|-x $$

Step-by-Step Solution

Answer

The graph of \( G(x) = |x| - x \) consists of a line \( y = -2x \) for \( x < 0 \) and \( y = 0 \) for \( x \geq 0 \).

Step 1: Understand the Function

The function given is \( G(x) = |x| - x \). This involves the absolute value function, \( |x| \), which affects the graph significantly. The absolute value returns the non-negative value of \( x \), meaning \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\).

Step 2: Determine Critical Points

Since \( G(x) \) involves an absolute value, the behavior changes at the point \( x = 0 \). We will calculate values both in the region where \(x \geq 0\) and \(x < 0\).

Step 3: Evaluate the Function for \( x \geq 0 \)

For \( x \geq 0 \), \( |x| = x \). Hence, the function becomes \( G(x) = x - x = 0 \). Therefore, for any non-negative \( x \), \( G(x) \) equals 0.

Step 4: Evaluate the Function for \( x < 0 \)

For \( x < 0 \), \( |x| = -x \). In this region, the function becomes \( G(x) = -x - x = -2x \). Here, \( G(x) \) would be positive because \( x \) is negative and multiplying by \(-2\) negates the sign.

Step 5: Create a Table of Values

Let's choose several values of \( x \) to substitute into the function:- For \( x = -2 \), \( G(-2) = -2(-2) = 4 \)- For \( x = -1 \), \( G(-1) = -2(-1) = 2 \)- For \( x = 0 \), \( G(0) = 0 \)- For \( x = 1 \), \( G(1) = 0 \)- For \( x = 2 \), \( G(2) = 0 \)This gives the points: (-2, 4), (-1, 2), (0, 0), (1, 0), (2, 0).

Step 6: Sketch the Graph

Plot the points from the table and connect them smoothly. The graph should show a line increasing from \( (x, y) = (-\infty, -\infty) \) reaching (0, 0) (since the function is \(-2x\) for negative \(x\)) and then lies along the x-axis (\(y=0\)) for \(x\geq 0\). Therefore, the graph forms a piecewise line that changes slope at \(x = 0\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function is crucial to grasping how a particular function will behave. In its simplest terms, the absolute value of a number is its distance from zero on the number line, which means it is always a non-negative number. For any real number \( x \), the absolute value \(|x|\) is defined as follows:

• If \( x \geq 0\), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

This definition results in the absolute value function having a distinctive V-shape when graphed. In the context of our function \( G(x) = |x| - x \), understanding where \(|x|\) shifts from \( x \) to \(-x\) (especially at \( x = 0 \)) is pivotal, as it will determine the behavior of the function in its different segments, ultimately affecting the graph's shape.

Analyzing Piecewise Functions

Piecewise functions are intriguing because they are made up of different expressions over different intervals. It's like having separate rules that apply depending on the value of \( x \). For example, in the function \( G(x) = |x| - x \), the absolute value creates a natural division:

• When \( x \geq 0 \), \( G(x) = x - x = 0 \).
• When \( x < 0 \), \( G(x) = -x - x = -2x \).

Effectively, \( G(x) \) is a piecewise function. For \( x \geq 0 \), the output is a horizontal line at \( y = 0 \), while for \( x < 0 \), the output is a line with a negative slope (moving upwards as \( x \) moves from negative to zero due to the \( -2x \) term). To sketch this graph, one should plot points from both cases and look for that seamless junction at \( x = 0 \), which is characteristic of such functions.

Creating and Using a Table of Values

A table of values is an excellent tool for graphing functions, particularly when trying to accurately capture function behavior across different intervals. It aids in visualizing how the function's output changes with respect to \( x \). To make a table of values for \( G(x) = |x| - x \), choose a range of \( x \) values from both significant intervals:

• For \( x \geq 0 \), use values such as 0, 1, 2.
• For \( x < 0 \), try values like -1, -2.

For example:

• \( x = -2 \), \( G(-2) = -2(-2) = 4 \)
• \( x = -1 \), \( G(-1) = -2(-1) = 2 \)
• \( x = 0 \), \( G(0) = 0 \)
• \( x = 1 \), \( G(1) = 0 \)
• \( x = 2 \), \( G(2) = 0 \)

By plotting these values, you get a set of points showing the graph rising from negative \( x \) values and levelling out at \( 0 \) for positive \( x \) values. This visualization makes it easier to draw the graph and understand the piecewise nature of \( G(x) \).