Complete the description of the piecewise function

Questions: Complete the description of the piecewise function graphed below. if -6 ≤ x ≤ -3 if -3 < x ≤ 1 f(x)=1 if 1 < x ≤ 6

Transcript text: Complete the description of the piecewise function graphed below. if $-6 \leq x \leq-3$ if $-3

Solution

Solution Steps

Step 1: Identify the segments of the piecewise function

The graph is divided into three segments:

From $ x = -6 $ to $ x = -3 $
From $ x = -3 $ to $ x = 1 $
From $ x = 1 $ to $ x = 6 $

Step 2: Determine the equations for each segment

First Segment ($ -6 \leq x \leq -3 $):
- The line passes through points (-6, 4) and (-3, -2).
- Slope $ m = \frac{-2 - 4}{-3 - (-6)} = \frac{-6}{3} = -2 $.
- Using point-slope form $ y - y_1 = m(x - x_1) $: \[ y - 4 = -2(x + 6) \implies y = -2x - 8 \]
Second Segment ($ -3 \leq x \leq 1 $):
- The line passes through points (-3, -2) and (1, -2).
- Slope $ m = \frac{-2 - (-2)}{1 - (-3)} = 0 $.
- This is a horizontal line at $ y = -2 $.
Third Segment ($ 1 \leq x \leq 6 $):
- The line passes through points (1, -2) and (6, 3).
- Slope $ m = \frac{3 - (-2)}{6 - 1} = \frac{5}{5} = 1 $.
- Using point-slope form $ y - y_1 = m(x - x_1) $: \[ y + 2 = 1(x - 1) \implies y = x - 3 \]

Step 3: Write the piecewise function

Combine the equations for each segment into a piecewise function: \[ f(x) = \begin{cases} -2x - 8 & \text{if } -6 \leq x \leq -3 \\ -2 & \text{if } -3 \leq x \leq 1 \\ x - 3 & \text{if } 1 \leq x \leq 6 \end{cases} \]

Final Answer

\[ f(x) = \begin{cases} -2x - 8 & \text{if } -6 \leq x \leq -3 \\ -2 & \text{if } -3 \leq x \leq 1 \\ x - 3 & \text{if } 1 \leq x \leq 6 \end{cases} \]

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Questions: Complete the description of the piecewise function graphed below. if -6 ≤ x ≤ -3 if -3 < x ≤ 1 f(x)=1 if 1 < x ≤ 6

Solution

Solution Steps

Final Answer

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