Question

sketch a graph of the piecewise - defined function.
f(x)=\begin{cases}x & \text{if }xleq0\\x + 4& \text{if }x>0end{cases}

Explanation:

Step1: Analyze the first - part of the function

For $y = x$ when $x\leq0$, it is a straight - line with slope $m = 1$ and $y$ - intercept $b = 0$. We draw the line $y=x$ for all $x$ values from negative infinity up to and including $x = 0$. The point $(0,0)$ is included, so we use a closed - circle at $(0,0)$.

Step2: Analyze the second - part of the function

For $y=x + 4$ when $x>0$, it is a straight - line with slope $m = 1$ and $y$ - intercept $b = 4$. We draw the line $y=x + 4$ for all $x$ values greater than $0$. The point $(0,4)$ is not included, so we use an open - circle at $(0,4)$.

To sketch the graph:

1. Plot the line $y = x$ for $x\leq0$. Mark the point $(0,0)$ with a closed - circle.
2. Plot the line $y=x + 4$ for $x>0$. Mark the point $(0,4)$ with an open - circle.

The graph has a break at $x = 0$.

Answer:

The graph consists of the line $y = x$ for $x\leq0$ (with a closed - circle at $(0,0)$) and the line $y=x + 4$ for $x>0$ (with an open - circle at $(0,4)$).