Questions: Graph the function. n(x)= -4 for -4<x<-2 x for -2 ≤ x<4 -x^2+16 for x ≥ 4

Transcript text: Graph the function. \[ n(x)=\left\{\begin{array}{ll} -4 & \text { for }-4

Solution

Solution Steps

Step 1: Understand the Piecewise Function

The given function \( n(x) \) is defined piecewise as follows:

\( n(x) = -4 \) for \( -4 < x < -2 \)
\( n(x) = x \) for \( -2 \leq x < 4 \)
\( n(x) = -x^2 + 16 \) for \( x \geq 4 \)

Step 2: Plot the First Piece

For \( -4 < x < -2 \), \( n(x) = -4 \). This is a horizontal line at \( y = -4 \) between \( x = -4 \) and \( x = -2 \) (not including the endpoints).

Step 3: Plot the Second Piece

For \( -2 \leq x < 4 \), \( n(x) = x \). This is a linear function with a slope of 1. Plot the line starting from \( (-2, -2) \) to \( (4, 4) \) (including \( x = -2 \) but not \( x = 4 \)).

Step 4: Plot the Third Piece

For \( x \geq 4 \), \( n(x) = -x^2 + 16 \). This is a downward-opening parabola starting from \( x = 4 \). Calculate a few points to plot:

At \( x = 4 \), \( n(4) = -4^2 + 16 = 0 \)
At \( x = 5 \), \( n(5) = -5^2 + 16 = -9 \)
At \( x = 6 \), \( n(6) = -6^2 + 16 = -20 \)

Final Answer

Graph the function by plotting the three pieces:

A horizontal line at \( y = -4 \) from \( x = -4 \) to \( x = -2 \) (open circles at endpoints).
A line with a slope of 1 from \( (-2, -2) \) to \( (4, 4) \) (closed circle at \( x = -2 \) and open circle at \( x = 4 \)).
A downward parabola starting at \( (4, 0) \) and continuing for \( x \geq 4 \).

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