Extrema found exactly.
Quadratics use completing the square. Higher-degree functions use calculus to set f'(x) = 0 and classify critical points.
Exact
AI Solver by AskSia
Range Solver: every possible output, graphed
Type or snap the function. AskSia finds extrema (using calculus or completing the square), analyzes end behavior, identifies horizontal asymptotes, and assembles the range as a union of intervals. The graph shows extrema, asymptotes, and any gaps in the output set.
Works with word problems, equations, code, and science prompts.
∫ 3x² · sin(x) dx
SubjectsCalculusAlgebraPhysicsChemistryBiologyCSStatisticsEcon
4.9 / 5 · trusted by 2M+ students · 50M+ problems solved
asksia.ai/solver
98% · Verified
Problem
Find the area enclosed between y = x² and y = 2x in the first quadrant.
1
Find the points of intersection
Set the curves equal: x² = 2x → x(x - 2) = 0, so x = 0 and x = 2.
2
Identify which curve is on top
On (0, 2), 2x > x², so y = 2x is the upper curve.
3
Set up the area integral
A = ∫₀² (2x - x²) dx
4
Integrate term-by-term
A = [ x² - x³/3 ]₀²
Final Answer
A = 4/3 sq. units
Quick Answer
What is the range of a function?
The range is the set of output values a function actually produces as the input varies over the domain. To find it, identify any extrema (highest and lowest points), analyze end behavior as x goes to plus and minus infinity, and check whether the function is continuous in between. For continuous functions on a connected domain, the range is the interval between the extreme values (or unbounded if no extremum exists in that direction). For discontinuous or piecewise functions, the range is a union of intervals.
98%
solution accuracy
50M+
problems solved
~1.5s
avg solve time
A+
study-ready explanations
Why AskSia Solver
Why students use AskSia for range analysis.
Extrema, asymptotes, and continuity all factored in.
End behavior tracked.
Limits as x goes to infinity are computed for polynomial, rational, exponential, and logarithmic functions.
Limit-aware
Asymptotes included.
Horizontal asymptotes set bounds on the range. AskSia identifies them and adjusts the output interval.
Asymptote-aware
Continuity considered.
Discontinuities create gaps in the range. AskSia detects them and writes the range as a union of intervals.
Continuity
Graph shows it.
The function plot has extrema, asymptotes, and the range highlighted on the y-axis as a visual confirmation.
Visual
Interval notation default.
Range appears as a union of intervals, with set-builder notation available on request.
Standard form
How It Works
Find any range in three steps.
Step 01
Enter the function.
Type, paste, snap a photo, or speak f(x). AskSia identifies the function type.
Input mode
Snap a Photo
Textbook, handwriting, screenshot
Paste Text
Word problem or equation
Calculator
LaTeX-ready equation editor
Step 02
AskSia analyzes behavior.
Extrema, end behavior, asymptotes, and continuity are each computed. Output values that the function cannot produce get excluded.
1
Set curves equal
x² = 2x → x = 0, x = 2
2
Set up the integral
A = ∫₀² (2x - x²) dx
3
Evaluate
A = [x² - x³/3]₀² = 4/3
Step 03
Read the range.
Final range in interval notation, with the graph showing extrema and asymptotes. Set-builder available on toggle.
Auto-generated diagram
Region between y = 2x and y = x² — area = 4/3
Available On
Solve anywhere
you study.
Every solve syncs across Web, iOS, and Android — start it at your desk, finish on your phone.
Web App
Full study studio
Split-panel interface with the worked solution on the left, the auto-generated diagram and AI tutor chat on the right.
Drag & drop image upload + LaTeX equation editor
Auto-generated diagrams render alongside steps
Side-panel AI tutor chat for hints and alt methods
Export to PDF, DOCX, Notion, or Google Docs
Hi! What are we studying today?
Ask about your homework, lecture, or readings...↑
Calculus
98% verified
1.4s
Step 4 of 4 · Evaluate
A = [x² - x³/3]₀² = 4/3
Mobile App
Snap & solve, anywhere
Open the camera, frame the problem, and the worked solution plus diagram appear in seconds.
One-tap snap-and-solve on iOS and Android
Pinch-to-zoom diagrams, swipe between steps
Auto-sync solves with your Web library
Offline review of saved solutions and flashcards
☰
AskSia
+What can I do for you?
Homework solver
Live transcribe
File summary
Snap
YouTube
Flashcard
Calc
98%
1.4s
Area between y=2x & y=x²
A = 4/3 sq. units ✓
Use Cases
Where range analysis matters.
Quadratic functions.
Range is [k, infinity) or (-infinity, k] depending on whether the parabola opens up or down. AskSia completes the square to find k.
Algebra 2
Polynomials of degree >= 3.
Range comes from extrema (found via calculus) and end behavior. AskSia handles cubics, quartics, and beyond.
Pre-Calc
Rational functions.
Horizontal asymptotes set bounds. AskSia checks whether the function crosses its asymptote.
Rational
Exponential and log.
Standard exponentials have range (0, infinity). Standard logs have range (-infinity, infinity). AskSia adjusts for shifts and reflections.
Logs
Trig functions.
Sine and cosine have range [-1, 1]. Tangent has all reals. AskSia adjusts for amplitude and vertical shifts.
Trig
Verify a stated range.
Paste a function and a proposed range. AskSia checks the bounds and any gaps.
Check
Compare
AskSia vs. ChatGPT,
Photomath & Symbolab.
General chatbots hallucinate. Photo solvers stop at math. AskSia is built for actual coursework with verified accuracy, visual learning, and every subject.
| Feature | AskSia Solver | ChatGPT | Photo Solvers |
|---|---|---|---|
| Solution accuracy | ✓ 98% | ~70-85%, hallucinations | ~90%, math only |
| Auto-generated diagrams | ✓ Every solve | Inconsistent / broken | Graphs only, math-only |
| Step-by-step explanations | ✓ Numbered + plain English | Inconsistent depth | ✓ Math steps |
| Subject coverage | ✓ Math, Physics, Chem, Bio, CS, Econ | ✓ Wide but unverified | Math only |
| Photo input | ✓ Handwriting + diagrams + code | Photos OK, weak on handwriting | ✓ Math photos only |
| Answer verification | ✓ Self-checked before display | No verification | Math engine only |
| Tutor follow-ups | ✓ Hints, alt methods, ELI5 | ✓ General chat | Not available |
| Practice and flashcards | ✓ One-tap from any solve | Manual prompting | Not available |
| Code debugging | ✓ Python, Java, C++, SQL... | ✓ Yes | Not available |
| Free to start | ✓ Daily solves, no card | Limited model access | Steps locked behind paywall |
FAQ
Frequently asked questions.
How is the range of a quadratic found?
Complete the square to rewrite f(x) = ax^2 + bx + c as a(x-h)^2 + k. The vertex is (h, k). If a > 0, the parabola opens up and the range is [k, infinity). If a < 0, it opens down and the range is (-infinity, k]. AskSia performs the completion of the square step by step and reads k off directly.
How does end behavior determine range?
For polynomials and rationals, end behavior tells you what y-values the function approaches as x goes to plus and minus infinity. If the function is unbounded above, the range extends to infinity in the positive direction. If bounded by a horizontal asymptote, that value is the limit. AskSia computes limits at infinity using leading-term analysis.
What about functions with horizontal asymptotes that the function crosses?
The horizontal asymptote describes only the limit at infinity. The function can cross its horizontal asymptote at finite x values. AskSia tests for crossings by setting f(x) equal to the asymptote and solving. If a crossing exists, the asymptote value is in the range (not just an unreached limit).
How is the range of a piecewise function handled?
Each piece is analyzed for its own range over its sub-domain. AskSia unions the resulting intervals, checking endpoints (open or closed) carefully, and removes any duplicates. The final range is reported as a union of intervals with the contributing piece labeled for each.
How accurate is AskSia?
AskSia is engineered for accuracy on standard high school and college coursework. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.
Can I get practice problems and flashcards?
Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.
How much does AskSia cost?
AskSia has a free plan that includes daily solves across all subjects. AskSia Pro and Super include unlimited solves, advanced subjects, the full AI tutor companion, exports, and priority response speed. See pricing for details.
Start Today
Every output, accounted for.
Join 2M+ students using AskSia to find function ranges with full step-by-step analysis. Extrema, asymptotes, interval notation, and a graph on every solve.
Let's Get in Touch
