Question

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

Answer 1

To determine if a given 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 first need to evaluate each piece of the piecewise function.

1. **For \( x < 1 \)**: The function is defined as \( f(x) = 2x + 6 \).
- At \( x = 1 \), \( f(1) = 2(1) + 6 = 8 \).

2. **For \( 1 \leq x < 6 \)**: The function is defined as \( f(x) = -\frac{1}{4}x - 3 \).
- At \( x = 1 \), \( f(1) = -\frac{1}{4}(1) - 3 = -\frac{1}{4} - 3 = -\frac{13}{4} = -3.25\).
- At \( x = 6 \), \( f(6) = -\frac{1}{4}(6) - 3 = -\frac{6}{4} - 3 = -\frac{3}{2} - 3 = -4.5 \).

3. **For \( x \geq 6 \)**: The function is defined as \( f(x) = 4 + 4x \).
- At \( x = 6 \), \( f(6) = 4 + 4(6) = 4 + 24 = 28 \).

Now, we confirm that the graph should:
- Approach 8 as \( x \) approaches \( 1 \) from the left (but is not defined at \( x = 1 \)).
- Start at \( -3.25 \) when \( x = 1 \) and go upwards to \( -4.5 \) at \( x = 6 \).
- Jump to \( 28 \) at \( x = 6 \) when continuing to \( 6 \) or above.

With this information, you can check if the graph shows these transitions correctly and reflects the behavior of the function given above.

If the graph does not show these behaviors (correct ranges and values), then it is not the correct graph for the function \(f(x)\).