Question

a piecewise function g(x) is represented by the graph. which functions represent a piece of the function? select three options. g(x) = -2x, -2 < x < 0; g(x) = -2, x < -2; g(x) = x - 2, -2 < x < 1; g(x) = -2x + 6, x ≥ 1; g(x) = (x/2) + 1, -2 ≤ x < 1

Explanation:

Step1: Analyze the leftmost piece

The leftmost part of the graph is a horizontal line at \( y = -2 \) for \( x < -2 \). So \( g(x) = -2, x < -2 \) is a valid piece.

Step2: Analyze the middle piece

The middle piece goes from \( (-2, 0) \) (closed dot) to \( (1, 1.5) \) (open dot). Let's find its equation. The slope \( m=\frac{1 - 0}{1 - (-2)}=\frac{1}{3}\)? Wait, no, wait the point at \( x=-2 \) is \( y = 0 \) (closed dot), and at \( x = 1 \), the open dot is at \( y=\frac{1}{2}+1 = 1.5 \)? Wait, no, let's check the option \( g(x)=\frac{x}{2}+1,-2\leq x < 1 \). When \( x=-2 \), \( y=\frac{-2}{2}+1=-1 + 1=0 \) (matches the closed dot at \( (-2,0) \)). When \( x = 1 \), \( y=\frac{1}{2}+1=1.5 \), and the open dot at \( x = 1 \) is at \( y = 1.5 \) (from the graph, the open dot at \( x = 1 \) is above \( y = 1 \)). So this function fits. Also, check \( g(x)=-2, x < -2 \): the leftmost part is horizontal at \( y=-2 \) for \( x < -2 \), which matches.

Step3: Analyze the rightmost piece

The rightmost piece starts at \( (1, 4) \) (closed dot) and has a slope. Let's check \( g(x)=-2x + 6, x\geq1 \). When \( x = 1 \), \( y=-2(1)+6 = 4 \) (matches the closed dot at \( (1,4) \)). When \( x = 3 \), \( y=-2(3)+6 = 0 \), which matches the graph crossing the x - axis at \( x = 3 \). So \( g(x)=-2x + 6, x\geq1 \) is valid. Also, check the other options:

• \( g(x)=-2x,-2 < x < 0 \): At \( x=-2 \), \( y = 4 \), but the middle piece starts at \( y = 0 \) at \( x=-2 \), so this is wrong.
• \( g(x)=x - 2,-2 < x < 1 \): At \( x=-2 \), \( y=-4 \), which doesn't match the closed dot at \( (-2,0) \), so wrong.

Answer:

B. \( g(x) = -2, x < -2 \), E. \( g(x)=\frac{x}{2}+1,-2\leq x < 1 \), D. \( g(x)=-2x + 6, x\geq1 \) (Note: Using the option labels as per the original: the options are:

1. \( g(x) = -2x, -2 < x < 0 \)
2. \( g(x) = -2, x < -2 \)
3. \( g(x) = x - 2, -2 < x < 1 \)
4. \( g(x) = -2x + 6, x\geq1 \)
5. \( g(x)=\frac{x}{2}+1,-2\leq x < 1 \)

So the correct options are:

1. \( g(x) = -2, x < -2 \)

1. \( g(x) = -2x + 6, x\geq1 \)

1. \( g(x)=\frac{x}{2}+1,-2\leq x < 1 \)