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

Solution Steps

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 \( x \geq 4 \)

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).

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 \( x = 4 \)).

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