Question

graph each function. give the domain and range.
$f(x) = \

$$\begin{cases} x^2 + 4x & \\text{if } x < -3 \\\\ 2 & \\text{if } x \\geq -3 \\end{cases}$$

$
(there is a coordinate grid image here)
domain: \underline{\\(\mathbb{r}\\)}
range: \underline{\\( (-\infty, -3) \\)}

Explanation:

Step1: Analyze domain of piecewise function

The function is defined for all real numbers: $x < -3$ and $x \geq -3$ cover every real value.

Step2: Analyze first piece ($x < -3$)

Rewrite $x^2 + 4x$ as $(x+2)^2 - 4$. This is a parabola opening upward, vertex at $(-2, -4)$. For $x < -3$, substitute $x=-3$: $(-3)^2 + 4(-3) = 9 - 12 = -3$. As $x \to -\infty$, $x^2 + 4x \to +\infty$, so this piece outputs $(-3, +\infty)$.

Step3: Analyze second piece ($x \geq -3$)

This is a constant function $y=2$, so it outputs only the value 2.

Step4: Combine ranges

Combine the outputs of both pieces: $(-3, +\infty) \cup \{2\}$. Since $2$ is already within $(-3, +\infty)$, the full range is $(-3, +\infty)$.

Answer:

Domain: All real numbers ($\mathbb{R}$)
Range: $(-3, +\infty)$