Problem 25 - 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

For \(x < 0\), \(G(x) = 0\) is a horizontal line; for \(x \geq 0\), \(G(x) = 2x\) increases with slope 2.

Step 1: Understand the Function

The given function to graph is \( G(x) = |x| + x \). This function combines the absolute value of \(x\) with \(x\) itself.

Step 2: Determine Critical Points

The critical point for this function occurs where \( |x| \) changes behavior, which is at \( x = 0 \). For values of \(x > 0\), both \(|x|\) and \(x\) are simply \(x\). For \(x < 0\), \(|x| = -x\).

Step 3: Make a Table for \( x \geq 0 \)

For \(x \geq 0\), \( |x| = x \). Therefore, \( G(x) = x + x = 2x \). Let's compute some values:- \(x = 0: G(x) = 2 \times 0 = 0\)- \(x = 1: G(x) = 2 \times 1 = 2\)- \(x = 2: G(x) = 2 \times 2 = 4\)

Step 4: Make a Table for \( x < 0 \)

For \(x < 0\), \( |x| = -x \). Therefore, \( G(x) = -x + x = 0 \). Let's compute a couple of values:- \(x = -1: G(x) = 0\)- \(x = -2: G(x) = 0\)

Step 5: Plotting the Points

Using the table values, plot the points on the graph:- For \(x \geq 0\): \((0,0), (1,2), (2,4)\) making a line with slope 2 passing through the origin.- For \(x < 0\): \((-1,0), (-2,0)\), make a horizontal line along \(y = 0\).

Step 6: Sketch the Graph

Draw the graph based on the plotted points. The graph is a line starting from (0,0), going upwards with a slope of 2 for \(x\geq 0\), and a horizontal line along the \(x\)-axis for \(x<0\).

Key Concepts

Exploring Absolute Value Functions

Absolute value functions form a special category of mathematical functions. They involve the absolute value, denoted by the expression \, which indicates the function's distance from zero on a number line. This aspect of absolute value functions eliminates any negative outcome, essentially converting any negative input into its positive equivalent. This characteristic can significantly alter the function's graph.

In the function \(G(x) = |x| + x\), the absolute value part converts any negative \(x\) to positive, while the addition of \(x\) ensures that positive values are amplified. Regardless of the \(x\) value, this function effectively transforms negative input to zero because \(-x+x=0\). For positive numbers, the output is twice the value of \(x\), as \(x+x=2x\). Thus, absolute value functions play a crucial role in determining the transformation and values output by a function, establishing distinct graph characteristics, such as non-negative outputs for all \(x\).

Understanding Piecewise Functions

The function \(G(x) = |x| + x\) is a prime example of a piecewise function. Piecewise functions are those defined by multiple sub-functions, each applying to a different interval of the main function's domain. For \(G(x)\), this results from combining the absolute value with a simple linear term.

- On the interval \(x \geq 0\), \(G(x) = 2x\), resulting in a simple line with a slope of 2.- On the interval \(x < 0\), \(G(x) = 0\), forming a horizontal line along the \(x\)-axis.

This dual behavior exemplifies why piecewise functions require careful analysis of each segment. Every sub-function plays a distinct role and affects the overall graph differently. Thus, understanding each piece is key to constructing the graph correctly and interpreting the function's behavior over different domains.

Identifying Critical Points in Graphs

Critical points in the context of graphing functions are points where your function's behavior changes. These include points where the derivative is zero or undefined. However, in the function \(G(x) = |x| + x\), the critical point arises due to the absolute value aspect, which causes the function to exhibit different properties.

The critical point here occurs at \(x = 0\), the point where the behavior shifts from horizontal to a sloping line as \(x\) changes from negative to non-negative. At \(x = 0\), the function transitions from a line that remains flat at zero to a line that rises through the plot as \(x\) becomes more positive. Recognizing such points helps in sketching the graph accurately and foreseeing how the function output changes in response to differing \(x\) values.