Question

f(x)=\

$$\begin{cases}-3 &\\text{if }x < - 5 \\\\x + 2 &\\text{if }-5\\leq x\\leq15 \\\\-x + 2 &\\text{if }x>15\\end{cases}$$

a.

Explanation:

Step1: Analyze \(x < - 5\)

For \(f(x)=-3\) when \(x < - 5\), it is a horizontal - line \(y = - 3\) with an open - circle at \(x=-5\) since the inequality is strict.

Step2: Analyze \(-5\leq x\leq15\)

For \(f(x)=x + 2\) when \(-5\leq x\leq15\). When \(x=-5\), \(y=-5 + 2=-3\); when \(x = 15\), \(y=15 + 2=17\). This is a line segment with endpoints \((-5,-3)\) (closed - circle) and \((15,17)\) (closed - circle).

Step3: Analyze \(x>15\)

For \(f(x)=-x + 2\) when \(x>15\). When \(x = 15\), \(y=-15 + 2=-13\). The line \(y=-x + 2\) starts with an open - circle at \(x = 15\) and has a slope of \(-1\).

Answer:

The correct graph is the one that shows a horizontal line \(y=-3\) for \(x < - 5\) (open - circle at \(x=-5\)), a line segment from \((-5,-3)\) to \((15,17)\) for \(-5\leq x\leq15\) (both endpoints closed - circles), and a line with slope \(-1\) starting with an open - circle at \(x = 15\) for \(x>15\). Without seeing the exact details of each option a, b, c, d, you can match based on the above - described characteristics. If we assume the options follow the standard graphing conventions, you need to check for these three parts: the horizontal part, the line - segment part, and the slanted part with the correct endpoints (open or closed).