Function study, part 1: domain and symmetries

The first mistake in a function study is almost never in the derivative, not in the sign, not in the asymptote. It is in the domain, calculated at minute zero, before you have even written anything interesting. And if you get it wrong there, every later calculation is invalidated: you can differentiate perfectly a function that does not exist at that x, find an asymptote at a point excluded from the domain, draw a graph that passes where it shouldn’t. The domain is not a bureaucratic stamp to put at the top of the page: it is the map of the territories where you have permission to reason. The whole rest of the study moves inside those borders, never outside.

This is the first chapter of a six-part series on how to tackle the complete guide to function study. Here we stop on just two things, domain and symmetries, but we cover them in full: it is the part that, if solid, saves you time for the whole rest of the test.

What the domain really is

The domain of a function $f$ is the set of $x$ for which the expression defining $f(x)$ makes sense — that is, produces a real number, not an impossible operation. Nothing else.

It is worth distinguishing two cases that students often confuse:

• Natural domain: the one you get by analysing only the mathematical expression, without knowing where it comes from. If $f(x)=\frac{1}{x-3}$, the natural domain is $x\ne 3$, full stop.
• Domain imposed by the problem: when the function models something concrete, the context adds conditions that the expression alone would not see. If $A(x)=x(10-x)$ is the area of a rectangle with sides $x$ and $10-x$, the natural domain would be all of $\mathbb{R}$, but physically you need $x>0$ and $10-x>0$, that is $0<x<10$. A negative side does not exist, even if algebraically the product can be calculated.

In pure analysis exercises you almost always work on the natural domain. In applied maximum and minimum problems (geometry, optimisation, physics) the imposed domain is often the most neglected part, and indeed it is the one that loses the most marks in tests: you find the maximum of a function forgetting that half of its natural domain makes no sense in the real problem.

The conditions of existence, one by one

Every “suspicious operation” inside the expression of $f$ generates a condition. We go through them all, with a worked example for each.

Denominator different from zero. If $x$ appears in a denominator, that denominator cannot vanish.

Factorise:

Condition: $x\ne 2\wedge x\ne 3$.

Beware of hidden denominators, those inside another fraction or inside a composite expression:

Here there are two denominators, not one: the inner one, $x\ne 0$, and the outer one, which vanishes when

that is $x=1$. If you stop at the “big” denominator and ignore the one inside, you lose half the domain.

Even-index root: argument greater than or equal to zero.

Condition: $4-x^2\ge 0⟹x^2\le 4⟹-2\le x\le 2$.

For an odd index (cube root, fifth root, …) there is no condition: $\sqrt[3]{x-1}$ exists for every real $x$, because the odd root of a negative number exists (it is negative). It is one of the most frequent mistakes: putting the condition $\ge 0$ on a cube root too.

Logarithm: argument strictly greater than zero.

Condition:

If the base is also a variable, as in ${\log }_x(5)$, two more conditions on the base are needed: $x>0$ and $x\ne 1$ (the base of a logarithm must be positive and different from 1, otherwise the operation is not defined or is trivial).

Tangent and cotangent: where cosine and sine vanish.

does not exist where $\cos x=0$, that is $x\ne \frac{\pi }{2}+k\pi$. The cotangent,

does not exist where $\sin x=0$, that is $x\ne k\pi$.

Arcsine and arccosine: argument between $-1$ and $1$.

Condition:

Non-integer exponent, or variable exponent: strictly positive base. When you have $f(x{)}^{g(x)}$ and $g(x)$ is not a fixed integer (or is even variable itself), the base must be positive because raising to a power with an arbitrary real exponent is defined only there.

Condition:

$x-3\ne 0$ is not enough: with a non-integer exponent, a negative base does not even make sense (think of $(-1{)}^{1/2}$).

Absolute value: no condition. I always clarify this because it generates a constant misunderstanding: $|x-2|$ is defined for every real $x$, it imposes nothing on the domain. The absolute value is an operation that can always be calculated, like a sum or a product. A condition arises only if the absolute value sits, for example, in a denominator or inside a logarithm — but it is that other operation that generates it, not the modulus itself.

How to put the conditions together: system, not union

When in a function there are several conditions at once, they must be put in a system, that is intersected, because all of them must be true at the same point $x$ for $f(x)$ to exist there. Using the union instead of the intersection is probably the most widespread mistake on this topic: taking the union widens the domain instead of narrowing it, and makes you include points where the function does not in fact exist.

Let’s take an example with three conditions together:

The conditions to impose are three:

1. The denominator does not vanish: $\ln (x-1)\ne 0⟹x-1\ne 1⟹x\ne 2$.
2. The argument of the root is non-negative: $x+2\ge 0⟹x\ge -2$.
3. The argument of the logarithm is positive: $x-1>0⟹x>1$.

I put them in a system:

Intersecting the first two, $x>1$ absorbs $x\ge -2$ (it is more restrictive). What remains is $x>1$, from which the point $x=2$ must be removed. The domain, in set notation, is:

and as a union of intervals:

Note that the point $x=2$ is not excluded “for aesthetics”: if you forget it, later you calculate a value of $f$ that is really a disguised division by zero.

How to write the domain (and where people go wrong writing it)

Three things to keep in mind about the form:

• The final set must be written as a union of disjoint intervals, with round brackets at excluded points and square ones at included points. $(1,2)\cup (2,+\infty )$ is correct; $(1,+\infty )\setminus \{2\}$ is equivalent but less standard in high school tests — use the first form.
• Don’t confuse $\cup$ with $\cap$: you use the union only to write the final result (the pieces of domain obtained are in union with each other), not to calculate it from the conditions, which always stay in a system.
• When the denominator is a logarithm, the condition most often forgotten is the second one: you impose the argument positive and forget that this logarithm, besides having to be defined, must not even vanish. And $\ln (g(x))=0$ means $g(x)=1$, not $g(x)=0$ — that is where most students fall.

Symmetries: even, odd, neither

A function is even if $f(-x)=f(x)$ for every $x$ in the domain: the graph is symmetric about the $y$ axis. It is odd if $f(-x)=-f(x)$ for every $x$ in the domain: the graph is symmetric about the origin.

There is a requirement that almost everyone skips: for the test $f(-x)$ to make sense, the domain must be symmetric about the origin — that is, if $x_0$ belongs to the domain, $-x_0$ must belong to it too. If this does not happen, the question “is it even or odd?” does not even arise: the function can be neither, regardless of how the expression is made.

Example: $f(x)=\sqrt{x}$ has domain $x\ge 0$, which is not symmetric (for $x_0=4$ there is $\sqrt{4}$, but $\sqrt{-4}$ does not exist). It makes no sense to calculate $f(-x)$: the function is neither even nor odd, and you already know it from the domain, before even looking at the expression.

Three examples on symmetric domains:

Even. $f(x)=x^4-2x^2+1$, domain $\mathbb{R}$ (symmetric).

Even, graph symmetric about the $y$ axis.

Odd. $f(x)=x^3-x$, domain $\mathbb{R}$.

Odd, symmetric about the origin.

Neither even nor odd. $f(x)=x^2+x$, domain $\mathbb{R}$.

This is not equal to $f(x)$ (so it is not even) and not equal to $-f(x)=-x^2-x$ (so it is not odd). It is the most common case by far: as soon as a polynomial mixes even and odd powers without cancellations, it comes out neither-nor.

A couple of practical rules that speed up recognition without having to expand $f(-x)$ every time:

• Even powers of $x$ ($x^2,x^4,\dots$) and constants give even contributions; odd powers ($x,x^3,\dots$) give odd contributions.
• $\cos x$ and $\cosh x$ are even; $\sin x$, $\tan x$ and $\sinh x$ are odd.
• The product (or quotient) of an even function and an odd one is odd; even times even, or odd times odd, is even.

The reason it is worth recognising symmetry straight away is practical, not aesthetic: if a function is even or odd, you study the graph only for $x\ge 0$ and then reflect it (about the $y$ axis if even, about the origin if odd). It means half the calculations on sign, derivative and limits on the right, and the other half obtained for free by symmetry.

Periodicity

Looking for periodicity makes sense above all for trigonometric functions (or combinations of them), not in general — on a polynomial or a logarithm it makes no sense to ask, because they cannot be periodic (apart from the constant function). $\sin (kx)$ and $\cos (kx)$ have period $\frac{2\pi }{|k|}$; for example $\sin (3x)$ has period $\frac{2\pi }{3}$.

When you add periodic functions with different periods, the lowest common multiple of the single periods is always a period of the sum, if the periods are commensurable — and it is the actual one apart from exact cancellations between the components, a case you will almost never meet at school. $\sin x+\cos (2x)$ has periods $2\pi$ and $\pi$: the period of the sum is $2\pi$.

For the maturità periodicity rarely requires a formal proof: it is enough to recognise it (it is usually declared by the presence of sine, cosine or compositions of them) and, if it is needed for the study of the graph, restrict the analysis to an interval as long as the period.

Symmetry about a vertical line (bonus)

Besides symmetry about the $y$ axis or the origin, a function can be symmetric about any vertical line $x=a$, not necessarily the $y$ axis: the condition is $f(a+t)=f(a-t)$ for every $t$. It is the typical case of parabolas, whose axis of symmetry almost never passes through the origin: $f(x)=x^2-4x+3$ has its vertex at $x=2$, and indeed

It is not a topic that is needed very often outside the conics, but it is useful for recognising at a glance the axis of a parabolic graph or similar without calculating the derivative.

The mistakes I see most often

1. Using the union instead of the intersection when there are several conditions. The final domain is always the intersection of the single conditions, then written as a union of intervals.
2. Forgetting the hidden denominator inside a logarithm, a root or a fraction of fractions — you see only the “main” denominator and lose a condition.
3. Putting the condition $\ge 0$ on an odd-index root, which does not need it.
4. Concluding “even” or “odd” without checking that the domain is symmetric. If the domain is not symmetric about the origin, the function is neither even nor odd, and you can say so straight from the domain, without calculating anything.
5. Applying a standard limit or a property of the logarithm before having checked the domain, getting a result that holds at a point where the function does not even exist.

How we work on this in lessons

• We start from a catalogue of 15-20 composite functions (roots inside logarithms, logarithms inside roots, denominators with several factors) and solve them all as a system, with no shortcuts, until the reflex “which conditions does this operation generate” becomes automatic.
• For symmetries, we work on pairs of examples very similar to each other (one even, one that looks even but isn’t) to train the eye not to trust the look of the formula.
• Generally 1-2 lessons are enough to make the standard conditions of existence (denominator, root, logarithm) automatic. To get to handling composite cases with three or four conditions confidently, and non-trivial symmetries, a few more exercises are needed: realistically one more consolidation lesson.
• We work online with a graphics tablet, writing step by step as in the test: the sequence “I identify the operation → I write the condition → I put it in a system” must come out on its own, without having to think about it.

With domain and symmetries solid, the next natural step is part 2: intercepts and sign study, which is built exactly on top of the domain you have just found. For the general picture of the subject there is the maths page; if the domain is your weak point, write to me for a free 30-minute introductory call via the contact page.

The other parts of the series, if you need them out of order: part 3: limits and asymptotes, part 4: monotonicity, maxima and minima, part 5: concavity and inflection points, part 6: the graph and worked examples. To check where you stand on the whole chapter, there is the Year 5 test on function study. Derivatives, which come in from part 4, have a separate guide: derivatives explained properly. The limits and asymptotes from which the graph starts, seen from the test side, are in limits, continuity and asymptotes: the Year 5 test.