I keep tripping over domain and range, especially when the function can be simplified or when there’s a real-life story attached. My brain wants a checklist, but then I second-guess everything.
Backstory: in high school I basically memorized “no dividing by zero, no square roots of negatives,” and called it a day. That worked until I started seeing problems where the formula changes form or where the variable is something like time or tickets sold. Then I freeze and feel like I’m missing something obvious.
Example 1: f(x) = sqrt(9 − x^2). I think the domain should be [−3, 3] because that keeps the inside of the square root nonnegative. For the range, I’m saying [0, 3], because it looks like the top half of a circle of radius 3. But I’m not 100% sure about the endpoints. Is it definitely 0 and 3 included? I can reason my way there, but I’m worried I’m just pattern-matching “semicircle” and not doing it properly.
Example 2: g(x) = (x^2 − 1)/(x − 1). I know this simplifies to x + 1, but x ≠ 1 because the original had a zero denominator there. When I write the range, should I think of g as the line y = x + 1 with a hole at x = 1? If so, I want to say the range is all real numbers except 2, since that’s what you’d get at x = 1. That feels right, but also a bit weird because the simplified formula happily outputs 2 if I forget about the hole. What’s the clean, correct way to state the domain and range here without accidentally “fixing” the hole by simplifying?
Real-life style: suppose p(t) = 5t + 2 is the number of widgets sold t hours after opening, and the store is only open for 0 ≤ t ≤ 8. Also, p should really be a whole number, not a fraction. How do I write the domain and range clearly in a case like this? Do I say the domain is [0, 8] but then also say t is real or integer? And for the range, do I list the integer values only, or is there a standard way to write that? I also get confused about whether the endpoints should be included in the domain or range when they represent things like “opening time” or “closing time.”
Could someone walk me through a reliable way to decide domain and range in these situations, especially:
– figuring out whether endpoints are included without having to graph,
– handling simplified expressions that hide holes,
– and writing domains and ranges for discrete quantities in word problems?
I might be overthinking this, but I keep second-guessing my answers and I’d love a sanity check.
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3 Responses
Here’s a tidy checklist with friendly elbows. Start with algebraic constraints, then apply any real‑life constraints, then simplify if you like-but keep the original domain in your back pocket so you don’t accidentally heal a hole. Example 1: f(x) = sqrt(9 − x^2). For the domain, require 9 − x^2 ≥ 0, so −3 ≤ x ≤ 3; endpoints are allowed because the inside can be 0 and sqrt(0) is fine. For the range, note sqrt is increasing and 9 − x^2 takes values from 0 (at x = ±3) up to 9 (at x = 0), so f(x) ranges from 0 to 3 inclusively: domain [−3, 3], range [0, 3]. Example 2: g(x) = (x^2 − 1)/(x − 1). The domain is all real numbers except x = 1. You can simplify to x + 1 for x ≠ 1, but keep the restriction: g(x) = x + 1, x ≠ 1. For the range, imagine the line y = x + 1 with a hole where x = 1 would give y = 2; that y is not produced anywhere else, so the range is all real numbers except 2. (General tip: removing an x only removes its y from the range if no other x gives that same y.) Example 3: p(t) = 5t + 2, with the store open 0 ≤ t ≤ 8 and p meant to be a whole number. Be explicit about the modeling choice: if time is continuous, the domain is [0, 8] and the range is [2, 42] but not integers; if you truly mean counts at whole hours, write domain {0, 1, …, 8} and range {5t + 2 : t in {0, …, 8}} = {2, 7, 12, 17, 22, 27, 32, 37, 42}. Endpoints belong if the expression is defined there and the context includes them (your “0 ≤ t ≤ 8” does), and they’re excluded only when the problem states a strict inequality or the formula misbehaves there.
Quick checklist: write the “no-go” algebra rules first (denominator ≠ 0, even roots ≥ 0, logs > 0), add real-world bounds/integers, simplify if you want but keep any excluded x’s, include endpoints exactly when the inequality is ≤/≥ or the story says opening/closing counts, then take the y-values hit and remove any y from holes.
Examples: f(x)=√(9−x^2) ⇒ 9−x^2≥0 gives domain [−3,3], and y runs 0 to 3 so range [0,3]; g(x)=(x^2−1)/(x−1) = x+1 with x≠1 ⇒ domain ℝ∖{1}, range ℝ∖{2} (hole at (1,2)); p(t)=5t+2 with 0≤t≤8 and whole-hour counts ⇒ domain {0,1,…,8}, range {2,7,12,…,42} (if time is continuous instead, domain [0,8], range [2,42]).
Quick checklist: decide where the original formula is defined (radicands ≥ 0, denominators ≠ 0), then intersect with any story limits; include endpoints when the conditions allow equality (≥, ≤), exclude when they forbid it, and remember simplification can’t “restore” forbidden points.
So f(x)=sqrt(9−x^2) has domain [-3,3] and range [0,3]; g(x)=(x^2−1)/(x−1) has domain R\{1} and range R\{2} (the line y=x+1 with a hole at x=1, y=2); and if p(t)=5t+2 over whole hours 0≤t≤8, domain {0,1,…,8} and range {2,7,12,…,42}-think of it like a floor plan: you can repaint the walls (simplify), but a missing tile (the hole) is still missing.