$Function Analysis$

Function Analysis

Complete step-by-step function analysis: domain, symmetry, zeros, asymptotes, derivative, extrema, concavity, inflection points and the graph. Polynomials and rational functions.

Function:

f(x) =

A polynomial or a ratio of polynomials: x^3 − 3x, (x^2 + 1)/x, x/(x^2 − 4), x^4 − 2x^2. The multiplication sign can be omitted: 2x, 3(x − 1).

Test yourself

The function analysis calculator performs a complete analysis of a polynomial or a rational function following the standard school plan: it finds the domain, symmetry, zeros, asymptotes, the derivative, intervals of increase and decrease and the extrema, the second derivative, concavity and inflection points, and finally plots the graph. Roots are found exactly when possible (for example, x = (1 + √5)/2), the rest approximately.

The plan of a function analysis

Domain D(f). For a fraction, exclude the points where the denominator is zero.
Even or odd. If f(−x) = f(x), the function is even (the graph is symmetric about the y-axis); if f(−x) = −f(x), it is odd (symmetric about the origin).
Intercepts. The x-intercepts are the solutions of f(x) = 0, the y-intercept is the point (0, f(0)).
Asymptotes. Vertical ones are at the zeros of the denominator; a horizontal or slant asymptote y = kx + b describes the behavior as x → ±∞.
Derivative f′(x). The zeros of the derivative are the critical points. Where f′(x) > 0 the function increases, where f′(x) < 0 it decreases. If the sign changes from + to − at a point, it is a maximum; from − to +, a minimum.
Second derivative f″(x). Where f″(x) > 0 the graph is concave up (∪), where f″(x) < 0 it is concave down (∩). A point where f″ changes sign is an inflection point.
The graph is drawn through all the points and asymptotes found.

Derivative of a quotient

(P / Q)′ = (P′ · Q − P · Q′) / Q²

The denominator Q² is positive on the whole domain, so the sign of the derivative is the sign of its numerator.

Asymptotes of a rational function

Degrees of the numerator and denominator	Asymptote
deg P < deg Q	horizontal y = 0
deg P = deg Q	horizontal y = (leading coefficient of P) / (leading coefficient of Q)
deg P = deg Q + 1	slant y = kx + b — the quotient of P divided by Q
deg P ≥ deg Q + 2	no horizontal or slant asymptote

If the numerator and the denominator are both zero at the same point, after canceling the graph has a hole there, not an asymptote: (x² − 1)/(x − 1) = x + 1 for x ≠ 1.

Example. f(x) = x³ − 3x.
D(f) = (−∞, +∞), the function is odd, zeros: x = 0, x = ±√3.
f′(x) = 3x² − 3 = 0 ⇒ x = ±1; x = −1 is a maximum, f(−1) = 2; x = 1 is a minimum, f(1) = −2.
f″(x) = 6x: for x < 0 the graph is concave down, for x > 0 concave up, (0, 0) is an inflection point.

How to enter the function

Polynomials and ratios of polynomials: x^3 − 3x, (x^2 + 1)/x, x/(x^2 − 4), (x − 1)^2(x + 2). The multiplication sign can be omitted, exponents are whole numbers up to 12. For functions with roots, logarithms and sines, use the graphing calculator.