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: 1) A line segment/ray for $y=x+3$ including the point $(0,3)$ and extending left; 2) A ray for $y=2x$ starting at the open point $(0,0)$ and extending right.

Answer

Step 1: Analyze the first piece $y=x+3$ for $x \leq 0$: Identify it as a linear function with slope 1 and y-intercept 3. Plot the endpoint at $x=0$: $f(0)=0+3=3$, which is a closed dot at $(0,3)$. Plot one additional point (e.g., $x=-3$, $f(-3)=0$) and draw a straight line/ray from $(0,3)$ through this point, extending leftward. Step 2: Analyze the second piece $y=2x$ for $x > 0$: Identify it as a linear function with slope 2 passing through the origin. Plot the endpoint at $x=0$: $f(0)=0$, which is an open dot at $(0,0)$ (since $x>0$ does not include 0). Plot one additional point (e.g., $x=2$, $f(2)=4$) and draw a straight ray from the open dot $(0,0)$ through this point, extending rightward. Step 3: Combine the two parts on the same coordinate grid to form the complete piecewise function graph.