Question

graph the piecewise - defined function
$f(x)=

$$\begin{cases}-3 - x&\\text{ if }xleq2\\-6 + 2x&\\text{ if }x>2\\end{cases}$$

$

choose the correct graph below

a.

b.

c.

d.

Explanation:

Step1: Analyze \( f(x) = -3 - x \) for \( x \leq 2 \)

This is a linear function with slope \( -1 \) and y - intercept \( -3 \). When \( x = 2 \), \( f(2)=-3 - 2=-5 \). Since \( x\leq2 \), the point at \( x = 2 \) is a closed dot (because the inequality is inclusive). We can also find another point, for example, when \( x = 0 \), \( f(0)=-3-0 = - 3 \).

Step2: Analyze \( f(x)=-6 + 2x \) for \( x>2 \)

This is a linear function with slope \( 2 \) and y - intercept \( -6 \). When \( x = 2 \), \( f(2)=-6 + 2\times2=-2 \). Since \( x>2 \), the point at \( x = 2 \) is an open dot (because the inequality is exclusive). When \( x = 3 \), \( f(3)=-6+2\times3 = 0 \).

Now let's check the graphs:

• For the first part (\( x\leq2 \)): The line \( y=-3 - x \) should pass through points like \( (0,-3) \), \( (2,-5) \) (closed dot).
• For the second part (\( x > 2 \)): The line \( y=-6 + 2x \) should have an open dot at \( (2,-2) \) and pass through \( (3,0) \) etc.

Looking at the options, we can eliminate options that don't match these characteristics. For example, option C: The first part ( \( x\leq2 \)) has a closed dot at \( x = 2 \) with \( y=-5 \) (from \( y=-3 - 2=-5 \)) and the second part ( \( x>2 \)) has an open dot at \( x = 2 \) with \( y=-2 \) (from \( y=-6 + 2\times2=-2 \)) and the line \( y=-6 + 2x \) has a positive slope. This matches our analysis.

Answer:

C