Questions: f(x) = -1, x ≤ -4 2x + 2, -4 < x ≤ 0 3, x > 0

f(x) = -1, x ≤ -4 2x + 2, -4 < x ≤ 0 3, x > 0
Transcript text: 2) $f(x)=\left\{\begin{array}{ll}-1, & x \leq-4 \\ 2 x+2, & -40\end{array}\right.$
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Solution

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Solution Steps

Step 1: Understanding the Piecewise Function

The given function \( f(x) \) is a piecewise function defined as: \[ f(x) = \begin{cases} -1, & x \leq -4 \\ 2x + 2, & -4 < x \leq 0 \\ 3, & x > 0 \end{cases} \]

Step 2: Plotting the First Piece

For \( x \leq -4 \), \( f(x) = -1 \). This is a horizontal line at \( y = -1 \) for all \( x \leq -4 \).

Step 3: Plotting the Second Piece

For \( -4 < x \leq 0 \), \( f(x) = 2x + 2 \). This is a linear function with a slope of 2 and a y-intercept of 2. We need to plot this line segment from \( x = -4 \) to \( x = 0 \).

  • At \( x = -4 \), \( f(-4) = 2(-4) + 2 = -8 + 2 = -6 \).
  • At \( x = 0 \), \( f(0) = 2(0) + 2 = 2 \).
Step 4: Plotting the Third Piece

For \( x > 0 \), \( f(x) = 3 \). This is a horizontal line at \( y = 3 \) for all \( x > 0 \).

Final Answer

The graph of the piecewise function \( f(x) \) is as follows:

  1. A horizontal line at \( y = -1 \) for \( x \leq -4 \).
  2. A line segment from \( (-4, -6) \) to \( (0, 2) \).
  3. A horizontal line at \( y = 3 \) for \( x > 0 \).

Plot these pieces on the given coordinate plane to visualize the function.

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