QiLing Tools

🔭Function Analyzer - Domain, Range, Extrema, Asymptotes

Curve sketching means writing down the same list of properties for every function: domain, range, symmetry, period, intercepts, where it increases and decreases, local extrema, concavity and inflection points, and asymptotes. Type f(x) — for example x^3-3x, (x^2-1)/(x-1), tan(x), ln(x) or x^3/(x^2-1) — and the analyzer does all of it. It reads the domain restrictions from the expression (denominator ≠ 0, log argument > 0, radicand ≥ 0, cos ≠ 0 for tan) and locates the boundaries precisely; first and second derivatives come from automatic differentiation, so their sign changes give exact increasing and decreasing intervals, maxima and minima, concavity and inflection points; limits at the edges of the domain and at infinity identify vertical, horizontal and slant asymptotes and classify removable, jump and infinite discontinuities. Answers are shown exactly when possible (fractions, √3, π/2 + kπ), periodic functions are described over one period, and you get a sign chart plus a graph with every key point marked.

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How to use Function Analyzer

  1. 1Enter f(x), e.g. x^3-3x, x/(x^2+1) or tan(x), or pick an example
  2. 2Adjust the analysis window if needed (default −10 ≤ x ≤ 10); zeros and extrema are searched there
  3. 3Read the property table, the sign chart and the annotated graph, or open the function in the grapher

FAQ

How do you tell whether a function is even or odd?
First check that the domain is symmetric about 0 — if it isn’t (ln x, or (x²−1)/(x−1) with x ≠ 1), the function is neither. Otherwise compare f(−x) with f(x): equal means even (symmetric about the y-axis), equal to −f(x) means odd (symmetric about the origin), and anything else is neither. The analyzer checks hundreds of points and also checks that excluded points come in symmetric pairs.
How are increasing intervals and extrema found?
From the sign of the derivative: f′(x) > 0 means increasing and f′(x) < 0 means decreasing; where f′ changes from + to − there is a local maximum, and from − to + a local minimum. The derivative is computed exactly by automatic differentiation, sign changes are located by bisection and recognised as exact values such as −1, √3 or π/4. Corners like |x| at 0, where f′ does not exist but the direction changes, count as extrema too.
What kinds of asymptotes are there?
A vertical asymptote x = a appears where f(x) → ±∞, like x = 0 for 1/x or x = π/2 + kπ for tan x. A horizontal asymptote is a finite limit as x → ±∞, like y = 0 for e^x as x → −∞. A slant asymptote y = kx + b has k = lim f(x)/x and b = lim [f(x) − kx], like y = x for x³/(x²−1). One-sided limits at every excluded point also tell asymptotes apart from removable and jump discontinuities.
Why are the range and period called estimates?
Domain, derivative signs and limits have reliable algorithms, but the range and the period are inferred from dense sampling plus limits at the ends. For ordinary functions the answers are exact — the range of x/(x²+1) is [−1/2, 1/2] and the period of sin x + cos x is 2π — but unusual cases (an extreme value outside the window, an unusual period) may need a different window. Endpoints are shown as fractions or surds when recognised and to 6 significant figures otherwise.