Question

sketch a graph of the piecewise - defined function.
$f(x)=\

$$\begin{cases}3x + 4&\\text{if }x < - 1\\4 - x&\\text{if }x\\geq - 1\\end{cases}$$

$

Explanation:

Step1: Analyze the first - part function

For \(y = 3x + 4\) when \(x\lt - 1\). The slope \(m = 3\) and the \(y\) - intercept \(b = 4\). We find the value of the function at \(x=-1\) for this part: \(y=3\times(-1)+4=1\). But since \(x\lt - 1\), we use an open - circle at the point \((-1,1)\) for this line.

Step2: Analyze the second - part function

For \(y = 4 - x\) when \(x\geq - 1\). The slope \(m=-1\) and the \(y\) - intercept \(b = 4\). When \(x=-1\), \(y=4-(-1)=5\). We use a closed - circle at the point \((-1,5)\) for this line.

Step3: Sketch the lines

Draw the line \(y = 3x + 4\) for \(x\lt - 1\) (open - circle at \((-1,1)\)) and the line \(y = 4 - x\) for \(x\geq - 1\) (closed - circle at \((-1,5)\)).

Answer:

Sketch the line \(y = 3x+4\) with an open - circle at the point \((-1,1)\) for \(x\lt - 1\) and the line \(y = 4 - x\) with a closed - circle at the point \((-1,5)\) for \(x\geq - 1\).