Mathematics
Study of a function: from domain to graph
Write the function and get the whole study in the order it is done on paper, with exact values where they exist: √3 rather than 1.732, 2 − √3 rather than 0.268.
The order of a study, and why it is that order
A study of a function is a sequence in which each step prepares the next. The domain comes first because everything else happens inside it: a zero outside the domain is not a zero, and a limit is taken only where the domain ends. Symmetry comes second because it halves the work — for an even or odd function, what holds for positive x is known for negative x too.
Intersections and sign come next, and they already fix the rough shape: where the graph sits above the axis and where below. The limits at the ends of the domain say how it behaves there, and each infinite limit at a finite point is a vertical asymptote. At ±∞ a finite limit gives a horizontal asymptote; an infinite one may hide an oblique asymptote, y = mx + q, with m = lim f(x)/x and q = lim (f(x) − mx).
The derivatives refine the picture. Where f′ is positive the function increases, where it is negative it decreases, and a change of sign marks a maximum or a minimum. f″ does the same for the bending: positive where the graph is concave up, negative where concave down, and a change of sign is a point of inflection. Only at the end, with every point and every asymptote in hand, is the graph drawn.
Common mistakes
- Calling every zero of f′ a maximum or a minimum. What decides is the change of sign: x³ has f′(0) = 0 and keeps increasing, so 0 is a stationary point of inflection.
- Looking for turning points only where f′ = 0. At a corner or a cusp f′ does not exist, and |x| has its minimum exactly there.
- Forgetting the domain when stating where a function increases: 1/x decreases on (−∞, 0) and on (0, +∞), not on the whole line, since 1/(−1) < 1/1.
- Looking for an oblique asymptote where there is already a horizontal one on the same side: a function can have one or the other as x → +∞, never both.
Frequently asked questions
What are the steps of a study of a function?
Domain, symmetry, intersections with the axes, sign, limits at the ends of the domain and asymptotes, first derivative with increasing and decreasing intervals and turning points, second derivative with concavity and inflections, and finally the graph.
How do I tell a maximum from a minimum?
From the sign of f′ either side of the point. If it goes from positive to negative the function stops rising and starts falling: a maximum. From negative to positive: a minimum. If it does not change sign, the point is neither.
What is a point of inflection?
A point of the graph where the concavity changes, from up to down or the other way. It is found where f″ changes sign; if f′ is also zero there, it is a stationary point of inflection, with a horizontal tangent.
Why are some values shown with ≈?
The zeros and limits are found numerically and then recognised: a fraction, a multiple of π, a root such as 2 − √3 or a power of e is written exactly. A value that is none of these, such as the root of cos x = x, is shown as a decimal approximation.
Does it work with trigonometric functions?
Yes, but a periodic function has infinitely many zeros and turning points, so they are listed within a range, from −2π to 2π unless you choose another. The domain, the limits and the asymptotes are stated in full wherever the function allows it.
How this calculation works
Conditions of existence are read off the expression: a denominator ≠ 0, the argument of an even root ≥ 0, the argument of a logarithm > 0, the argument of arcsine and arccosine between −1 and 1, and cos u ≠ 0 for tan u. f′ and f″ are taken symbolically with the rules of differentiation. Zeros of f, f′ and f″ are found by bisection on a fine grid, and double zeros by minimising |f|. Limits are read off the values of f on the way to the point, where the steps stop shrinking before rounding error takes over. Vertical asymptote at x = a if a one-sided limit at a is infinite; horizontal asymptote y = L if lim x→±∞ f(x) = L; oblique asymptote y = mx + q with m = lim f(x)/x ≠ 0 and q = lim (f(x) − mx), both finite. A maximum where f′ goes from + to −, a minimum from − to +; an inflection where f″ changes sign.
Related calculators
Domain of a function
The conditions of existence and the domain written as intervals.
Limits
One-sided and two-sided limits, at a point or at infinity, with the indeterminate form.
Asymptotes
Vertical, horizontal and oblique asymptotes, with the limits that give them.
Derivatives
The derivative of a function and the rules used to get it.