Question

f(x) = \

$$\begin{cases} \\frac{1}{2}x + 6 &, -4 \\leq x < 0 \\\\ -5 &, 0 \\leq x \\leq 7 \\end{cases}$$

what is the graph of f?
choose 1 answer:
a (with graph)
b (with graph)

Explanation:

Step1: Analyze the first piece of the function

The first piece is \( f(x)=\frac{1}{2}x + 6 \) for \( - 4\leq x<0 \). Let's find the endpoints. When \( x=-4 \), \( f(-4)=\frac{1}{2}(-4)+6=-2 + 6 = 4 \). When \( x = 0 \), since the domain is \( x<0 \) for this piece, we use an open circle at \( x = 0 \). The slope is \( \frac{1}{2} \), so it's a line with positive slope from \( (-4,4) \) to an open circle at \( (0,6) \) (because \( f(0)=\frac{1}{2}(0)+6 = 6 \) but not included here).

Step2: Analyze the second piece of the function

The second piece is \( f(x)=-5 \) for \( 0\leq x\leq7 \). This is a horizontal line. At \( x = 0 \), we have a closed circle (since \( x = 0 \) is included in this domain) at \( (0,-5) \), and it goes to \( (7,-5) \) with a closed circle at \( x = 7 \).

Now let's check the options:

• For option A: At \( x = 0 \), the first piece has a closed circle (which is wrong, should be open) and the second piece has an open circle (wrong, should be closed).
• For option B: The first piece has an open circle at \( x = 0 \) (correct) and the second piece has a closed circle at \( x = 0 \) (correct), and the points match the function values.

Answer:

B