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QUESTION IMAGE

question graph the following function on the axes provided. $f(x)=\\beg…

Question

question graph the following function on the axes provided. $f(x)=\

$$\begin{cases}-1&\\text{for}&-4

$ click and drag to make a line. click the line to delete it. click on an endpoint of a line to change it.

Explanation:

Step1: Graph the first piece

For \( -4 < x < 1 \), \( f(x) = -1 \). This is a horizontal line segment. The endpoints are open circles at \( x = -4 \) (since \( x > -4 \)) and \( x = 1 \) (since \( x < 1 \)), with \( y = -1 \) for all \( x \) in this interval. So we draw a horizontal line from \( x = -4 \) (open circle) to \( x = 1 \) (open circle) at \( y = -1 \).

Step2: Graph the second piece

For \( 1 < x \leq 5 \), \( f(x) = x + 2 \). This is a linear function. We find two points:

  • When \( x = 1 \) (but \( x > 1 \), so open circle), \( f(1) = 1 + 2 = 3 \) (open circle at \( (1, 3) \)).
  • When \( x = 5 \), \( f(5) = 5 + 2 = 7 \) (closed circle at \( (5, 7) \)).

We also can find another point, e.g., when \( x = 2 \), \( f(2) = 2 + 2 = 4 \) (point \( (2, 4) \)). Then we draw a line from the open circle at \( (1, 3) \) through \( (2, 4) \) to the closed circle at \( (5, 7) \).

Answer:

The graph consists of a horizontal line segment \( y = -1 \) from \( x = -4 \) (open) to \( x = 1 \) (open), and a line segment \( y = x + 2 \) from \( x = 1 \) (open) to \( x = 5 \) (closed).