Two rays and one line segment are plotted on a coordinate plane.

Question

Is this the correct graph for the piecewise function f(x)=⎧⎩⎨⎪⎪2x+6 if x<1−14x−3 if 1≤x<64+4x if x≥6
f
(
x
)
=
{
2
x
+
6

if

x
<
1
−
1
4
x
−
3

if

1
≤
x
<
6
4
+
4
x

if

x
≥
6
?

Enter 1 for yes or 2 for no.

(1 point)

Answer 1

To determine if the provided graph accurately represents the piecewise function

\[ f(x) = \begin{cases} 2x + 6 & \text{if } x < 1 \ -\frac{1}{4}x - 3 & \text{if } 1 \leq x < 6 \ 4 + 4x & \text{if } x \geq 6 \end{cases} \]

we need to analyze the function segments:

1. For \( x < 1 \): The function is a ray represented by the equation \( f(x) = 2x + 6 \). This line has a positive slope and should approach \( f(1) \) from below as \( x \) approaches 1.

2. For \( 1 \leq x < 6 \): The function is described by \( f(x) = -\frac{1}{4}x - 3 \). This is a decreasing line segment starting at \( x = 1 \) (where the value should equal \( f(1) \)) and goes until \( x = 6 \).

3. For \( x \geq 6 \): The function is defined by \( f(x) = 4 + 4x \). This part of the piecewise function will start at the point where \( x = 6 \) and continues as a ray with a positive slope.

If the graph matches these characteristics—specifically showing one ray for the first condition, a line segment for the second condition, and another ray for the last condition—it would be correct. If any conditions are violated (like the continuity at x = 1 and x = 6), the graph would not accurately reflect the piecewise function.

Please carefully analyze the plotted graph against the descriptions above to make your decision. If the graph fulfills all the criteria laid out, respond with 1 (yes). Otherwise, respond with 2 (no).

Since I cannot see the graph, I cannot determine the correctness directly. Please use the description to analyze the graph visually. If everything aligns with the piecewise definition, the answer is 1, else it is 2.