Algebra 2 worksheet focusing on domains, ranges, and end behavior with function diagrams and interval notation exercises.

Let’s work through the problems step by step.

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Problem 1: Are the following relations a function? Why or why not?

A function is a special kind of relation where each input (x-value) has exactly one output (y-value).
In diagrams, if any x-value points to more than one y-value, it’s NOT a function.

We’ll go one by one:

(a) Left oval: inputs are -2, -1, 0, 1, 2 → all map to only ONE output each → ✔ Function
(b) Input “-3” maps to BOTH 1 and 4 → ✘ Not a function
(c) All inputs (-5, -2, 0, 3, 6) have only one arrow out → ✔ Function
(d) Input “-2” maps to both -3 and 1 → ✘ Not a function
(e) Graph: passes vertical line test — no vertical line hits more than one point → ✔ Function
(f) Graph: same as above — every x has only one y → ✔ Function
(g) Graph: looks like a sideways parabola — at some x-values (like x=1), there are two y-values → ✘ Not a function
(h) Graph: again, for some x (say x=2), two y-values exist → ✘ Not a function

✔ So answers:
(a) Yes
(b) No
(c) Yes
(d) No
(e) Yes
(f) Yes
(g) No
(h) No

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Problem 2: Write intervals shown on number lines using inequality notation, set notation, and interval notation.

Recall:
- Open circle = not included → use < or >
- Closed circle = included → use ≤ or ≥
- Parentheses ( ) = open end
- Brackets [ ] = closed end

(a) Number line: from -4 (open) to 3 (closed)

→ Inequality: -4 < x ≤ 3
→ Set notation: {x | -4 < x ≤ 3}
→ Interval: (-4, 3]

(b) From -1 (closed) to ∞ (always open)

→ Inequality: x ≥ -1
→ Set: {x | x ≥ -1}
→ Interval: [-1, ∞)

(c) From -∞ to 4 (open)

→ Inequality: x < 4
→ Set: {x | x < 4}
→ Interval: (-∞, 4)

(d) Two separate parts: from -∞ to -3 (closed), and from 6 (open) to ∞

→ Inequality: x ≤ -3 OR x > 6
→ Set: {x | x ≤ -3 or x > 6}
→ Interval: (-∞, -3] ∪ (6, ∞)

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Final Answer:

1. (a) Yes (b) No (c) Yes (d) No (e) Yes (f) Yes (g) No (h) No
2. (a) Inequality: -4 < x ≤ 3; Set: {x | -4 < x ≤ 3}; Interval: (-4, 3]
(b) Inequality: x ≥ -1; Set: {x | x ≥ -1}; Interval: [-1, ∞)
(c) Inequality: x < 4; Set: {x | x < 4}; Interval: (-∞, 4)
(d) Inequality: x ≤ -3 or x > 6; Set: {x | x ≤ -3 or x > 6}; Interval: (-∞, -3] ∪ (6, ∞)