Question

carefully graph each of the following. identify whether or not the graph is a function. then, evaluate the graph at any specified domain value. you may use your calculators to help you graph, but you must sketch it carefully on the grid!

1. \\( f(x) = \

$$\begin{cases} x+5 & x < -2 \\\\ -2x-1 & x \\ge -2 \\end{cases}$$

\\)

function? yes or no

\\( f(3) = \\)

\\( f(-4) = \\)

\\( f(-2) = \\)

Explanation:

Identify the piecewise function rules

The piecewise function is defined as:

$$ f(x) =
LATEXBLOCK0
$$

Evaluate the function at specified values

Evaluate \(f(-2)\):
Since \(x = -2\) satisfies \(x \ge -2\), use the second rule:

$$ f(-2) = -2(-2) - 1 = 4 - 1 = 3 $$

Evaluate \(f(-4)\):
Since \(x = -4\) satisfies \(x < -2\), use the first rule:

$$ f(-4) = -4 + 5 = 1 $$

Evaluate \(f(3)\):
Since \(x = 3\) satisfies \(x \ge -2\), use the second rule:

$$ f(3) = -2(3) - 1 = -6 - 1 = -7 $$

Determine if the graph is a function

A relation is a function if each input \(x\) maps to exactly one output \(y\).
Since the domains \(x < -2\) and \(x \ge -2\) do not overlap, each \(x\) has exactly one value.
Thus, it passes the vertical line test.
The relation is a function: Yes.

Determine key points for graphing

For the first piece \(y = x + 5\) when \(x < -2\):

• As \(x \to -2\) from the left, \(y \to 3\) (open circle at \((-2, 3)\)).
• At \(x = -4\), \(y = 1\).
• At \(x = -5\), \(y = 0\).

For the second piece \(y = -2x - 1\) when \(x \ge -2\):

• At \(x = -2\), \(y = 3\) (solid circle at \((-2, 3)\)).
• At \(x = 0\), \(y = -1\).
• At \(x = 3\), \(y = -7\).

Since both pieces meet at the point \((-2, 3)\), the graph is continuous at this boundary.

Answer:

• Function? Yes
• Evaluations:
• \(f(-2) = 3\)
• \(f(-4) = 1\)
• \(f(3) = -7\)