Problem 68 - Beginning and Intermediate Algebra

Question

Graph the following piecewise functions. $$g(x)=\left\\{\begin{aligned}x, & & x & \leq 0 \\\2 x+3, & & x>0\end{aligned}\right.$$

Step-by-Step Solution

Answer

To graph the piecewise function \(g(x) = \begin{cases} x, & x\leq0\\ 2x+3, & x>0 \end{cases}\), follow these steps: 1. Graph \(y = x\) for \(x\leq 0\), including the point (0,0), extending the line only towards negative x-values (left of the y-axis). 2. Graph \(y = 2x + 3\) for \(x > 0\), starting from the point (0,3) and extending the line only towards positive x-values (right of the y-axis). 3. Combine the two parts on the same coordinate plane to form the complete graph of the piecewise function g(x).

Step 1: Graph the first part of the piecewise function

For this part, we have $$g(x) = x$$ and we are considering the domain where $$x \leq 0$$. Since this is a simple linear function with a slope of 1 and a y-intercept of 0, we can graph it by drawing a straight line through the origin (0,0), extending it only towards negative x-values (to the left of y-axis).

Step 2: Graph the second part of the piecewise function

This portion of the piecewise function is defined as $$g(x) = 2x + 3$$ when $$x > 0$$. Another linear function with a slope of 2 and a y-intercept of 3, we can start by plotting the point (0, 3) on the y-axis, and draw a line with a slope of 2, ensuring that the line begins after the y-axis and does not include the y-axis.

Step 3: Combine the two parts to create the graph of the piecewise function

Combine the graphs of the two parts of the piecewise function by keeping both the lines together on the same coordinate plane. The final graph should look like this: - A straight line from the origin (0,0) with a slope of 1, extending only to the left of the y-axis, including the y-axis. - A straight line starting from the point (0, 3) with a slope of 2, extending only to the right of the y-axis, excluding the y-axis. These lines will form a "broken line" representing the function g(x) in its entirety.

Key Concepts

Linear Functions

Linear functions are one of the simplest and most fundamental types of functions in mathematics. They describe relationships where there is a constant rate of change between the independent variable and the dependent variable. In other words, the graph of a linear function is always a straight line. The general form of a linear function is given by the equation:\[ y = mx + b \]where:

• \( m \) is the slope of the line, representing the rate of change.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

For example, in the piecewise function given, the part where \( g(x) = x \) has a slope \( m = 1 \) and a y-intercept \( b = 0 \). The line is diagonal and passes through the origin. Similarly, for the part \( g(x) = 2x + 3 \), the slope \( m = 2 \) indicates that the line is steeper, and the line crosses the y-axis at the point (0, 3). This consistent rule helps you predict the direction and steepness of any linear function.

Graphing Functions

Graphing functions involves plotting them on a Cartesian coordinate system, where you can see the relationship between the x (independent variable) and y (dependent variable). For linear functions like in our piecewise function example, graphing is straightforward:

• Start by identifying the slope \( m \) and y-intercept \( b \).
• Plot the y-intercept on the y-axis.
• Use the slope to determine the next points. For example, a slope of 2 means "go up 2 units for every 1 unit right."
• Draw a straight line through the plotted points.

When dealing with piecewise functions, you'll graph each section according to its own rule and specified domain. You must ensure that each part of the piecewise function only appears within its given domain, and properly visualize any discontinuities or conditions where the function changes its definition. In our exercise example, the function is graphed in two parts, one where \( x \leq 0 \), and another where \( x > 0 \). Both parts create distinct segments that are combined to complete the graph.

Function Domains

The domain of a function describes all the possible input values (x-values) that the function can accept. For piecewise functions, different rules apply in different parts of the domain. In our example, the function \( g(x) \) is defined using two linear parts:

• For \( x \leq 0 \), \( g(x) = x \). The domain here includes all non-positive x-values, starting from negative values up to zero.
• For \( x > 0 \), \( g(x) = 2x + 3 \). This domain consists of all positive x-values, but does not include zero.

Understanding the domain of each part is crucial when graphing piecewise functions. Each segment of the graph reflects the defined domain, ensuring that no overlap or misrepresentation occurs. Domains help visualize where each part ends or begins on the x-axis, and they dictate how the graph should look.