Question

question 1
which of the following graphs represents the given piece - wise function?
$f(x)=\

$$\begin{cases}-3x - 4, & x < - 2\\\\\\frac{1}{2}x + 1, & x\\geq - 2\\end{cases}$$

$

Explanation:

Step1: Analyze the first - part of the piece - wise function

For \(y = - 3x - 4\) when \(x\lt - 2\). The slope \(m=-3\) and the \(y\) - intercept is \((0, - 4)\). When \(x=-2\), \(y=-3\times(-2)-4=6 - 4 = 2\). But since \(x\lt - 2\), the point \((-2,2)\) is not included in this part of the graph, so we have an open - circle at \((-2,2)\) for this line.

Step2: Analyze the second - part of the piece - wise function

For \(y=\frac{1}{2}x + 1\) when \(x\geq - 2\). The slope \(m = \frac{1}{2}\) and the \(y\) - intercept is \((0,1)\). When \(x=-2\), \(y=\frac{1}{2}\times(-2)+1=-1 + 1=0\). Since \(x\geq - 2\), the point \((-2,0)\) is included in this part of the graph, so we have a closed - circle at \((-2,0)\) for this line.

Answer:

The graph that has a line with slope \(-3\) and \(y\) - intercept \(-4\) with an open - circle at \((-2,2)\) for \(x\lt - 2\) and a line with slope \(\frac{1}{2}\) and \(y\) - intercept \(1\) with a closed - circle at \((-2,0)\) for \(x\geq - 2\). (Since no options are given, this is a description of the correct graph).