Question

1. choose the best answer. which is the algebraic definition of the piecewise function graph? options: $f(x)\

$$\begin{cases}y = 2x + 3&-2\\leq x < 0\\\\y = -3x + 3&0\\leq x < 2\\end{cases}$$

$, $f(x)\

$$\begin{cases}y = 2x + 3&-1\\leq x < 0\\\\y = -3x + 3&0\\leq x < -3\\end{cases}$$

$, $f(x)\

$$\begin{cases}y = -2x + 3&-2\\leq x < 0\\\\y = 3x + 3&0\\leq x < 2\\end{cases}$$

$, none of these

Explanation:

Step1: Analyze the first segment (-2 ≤ x < 0)

We need to find the equation of the line from x = -2 to x = 0. Let's take two points on this segment. From the graph, when x = -2, y = -1; when x = 0, y = 3. The slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-1)}{0 - (-2)} = \frac{4}{2} = 2 \). Using point-slope form with (0, 3), the equation is \( y - 3 = 2(x - 0) \), so \( y = 2x + 3 \). The domain here is -2 ≤ x < 0 (since at x = -2, the point is included, and at x = 0, we switch to the next segment).

Step2: Analyze the second segment (0 ≤ x < 2)

Now, for the segment from x = 0 to x = 2. Take two points: (0, 3) and (2, -3). The slope \( m = \frac{-3 - 3}{2 - 0} = \frac{-6}{2} = -3 \). Using point-slope form with (0, 3), the equation is \( y - 3 = -3(x - 0) \), so \( y = -3x + 3 \). The domain here is 0 ≤ x < 2 (at x = 0, it's included, and at x = 2, the point is not included as per the graph).

Step3: Compare with options

The first option has the equations \( y = 2x + 3 \) for -2 ≤ x < 0 and \( y = -3x + 3 \) for 0 ≤ x < 2, which matches our analysis. The second option has incorrect domains (e.g., 0 ≤ x < -3 is invalid), and the third option has an incorrect slope for the first segment.

Answer:

\( f(x)

$$\begin{cases}y = 2x + 3 & -2 \leq x < 0 \\ y = -3x + 3 & 0 \leq x < 2\end{cases}$$

\) (the first option)