Question

the function f is defined as follows.
f(x)=\begin{cases}3 + x&\text{if }x<0\\x^{2}&\text{if }xgeq0end{cases}
(a) find the domain of the function.
(b) locate any intercepts.
(c) graph the function.
(d) based on the graph, find the range.
(a) the domain of the function f is (square). (type your answer in interval notation.)

Explanation:

Step1: Determine domain

Function defined for all x < 0 and x ≥ 0, covering all real numbers.

Step2: Find x-intercepts

For x < 0: 3 + x = 0 ⇒ x = -3. For x ≥ 0: x² = 0 ⇒ x = 0. Thus (-3, 0), (0, 0).

Step3: Find y-intercept

x = 0: f(0) = 0² = 0 ⇒ (0, 0).

Step4: Analyze range from graph

x < 0: f(x) = 3 + x < 3; x ≥ 0: f(x) = x² ≥ 0. Combined, y ≥ 0.

Answer:

(a) (-∞, ∞)
(b) x-intercepts: (-3, 0), (0, 0); y-intercept: (0, 0)
(c) Graph: For x < 0, line with slope 1 through (0, 3) (open circle); for x ≥ 0, parabola x² starting at (0, 0) (closed circle).
(d) [0, ∞)