Which option represents the piecewise function of the graph

Question

On graph one line is on (0,0) and (2,4) but has an open circle on the (2,4)
Second line (2,0) and closed dot on (5,9)
Options
f(x)= 2x, 0<x<=2
3x-6, 2<x<=5

f(x)= 2x, 0<=x<2
3x-6, 2<=x<=5

f(x)= 2x, 0<=x<4
3x-6, 0<=x<9

f(x)= 1/2x, 0<=x<2
1/3+2, 2<=x<5

Answer 1

To analyze the graph based on the description you've provided:

1. The first line goes from (0,0) to (2,4) but has an open circle at (2,4), indicating that the endpoint is not included in this segment. This means the function is defined for \(0 \leq x < 2\), and the equation of the line can be determined as \(f(x) = 2x\).

2. The second line starts at (2,0) and extends to (5,9), indicated by a closed dot at (5,9), showing that the endpoint is included. This means the function is defined for \(2 < x \leq 5\), and the equation can be determined to be \(f(x) = 3x - 6\).

Combining these observations, the piecewise function can be represented as:

\[ f(x) = \begin{cases} 2x & \text{for } 0 \leq x < 2 \ 3x - 6 & \text{for } 2 < x \leq 5 \end{cases} \]

Looking through the options provided:

• \( f(x) = 2x, \quad 0 < x \leq 2 \) \
• \( f(x) = 2x, \quad 0 \leq x < 2 \) \
• \( f(x) = 2x, \quad 0 \leq x < 4 \) \
• \( f(x) = \frac{1}{2}x, \quad 0 \leq x < 2 \) \

Only the second option fits:

\[ f(x)= 2x, \quad 0 \leq x < 2 \ 3x-6, \quad 2 \leq x \leq 5 \]

Therefore, the correct choice is:

f(x)= 2x, \quad 0 \leq x < 2 \ 3x-6, \quad 2 < x \leq 5.