How to Graph the Piecewise Function $f(x)=\begin{cases} x+3, x \leq 0 \\ 2x, x 0 \end{cases}$

Question

Graph the piecewise function $f(x)=\begin{cases} x+3, & \text{if } x \leq 0 \\ 2x, & \text{if } x > 0 \end{cases}$

Accepted Answer

A piecewise graph with two segments: 1) A line segment/ray with slope 1, y-intercept 3, including the point (0,3) and extending left; 2) A ray with slope 2, starting at the open point (0,0) and extending right.

Answer

Step 1: Analyze the first piece $y = x + 3$ for $x \leq 0$. Calculate key points: when $x=0$, $y=3$ (closed dot, since $x=0$ is included); when $x=-3$, $y=0$. Plot these points and draw a straight line extending to the left from (0,3). Step 2: Analyze the second piece $y = 2x$ for $x > 0$. Calculate key points: when $x=0$, $y=0$ (open dot, since $x=0$ is not included); when $x=2$, $y=4$. Plot these points and draw a straight line extending to the right from the open dot at (0,0). Step 3: Combine the two segments on the same coordinate plane to form the complete graph of the piecewise function.