Question

graph the piece - wise defined function. f(x) = { 2x + 5 x ≤ 0; 1/3x + 1 x > 0. choose the correct graph below.

Explanation:

Step1: Analyze \(f(x)=2x + 5\) for \(x\leq0\)

When \(x = 0\), \(y=2\times0 + 5=5\). The slope is \(m = 2\). Since \(x\leq0\), we have a line - segment or ray starting from \((0,5)\) (closed - circle because \(x = 0\) is included) and going down to the left with slope \(2\).

Step2: Analyze \(f(x)=\frac{1}{3}x + 1\) for \(x>0\)

When \(x = 0\), \(y=\frac{1}{3}\times0+1 = 1\). But \(x>0\), so the graph starts at the point \((0,1)\) with an open - circle (because \(x = 0\) is not included) and has a slope of \(\frac{1}{3}\), going up to the right.

Answer:

(Without seeing the actual graphs A, B, C, we can't choose a specific option. But the correct graph should have a line \(y = 2x+5\) with a closed - circle at \((0,5)\) for \(x\leq0\) and a line \(y=\frac{1}{3}x + 1\) with an open - circle at \((0,1)\) for \(x>0\).)