Function study, part 2: intercepts and sign study

If I had to name the most underrated step in the whole function study, I would have no doubt: it is the sign study. Not because it is hard (it is almost always the most mechanical part of the whole procedure) but because most students compute it, write it on one line, and then never use it again. They get to the final graph and redraw everything by intuition, ignoring that they already held the information telling them exactly where the graph could pass and where it couldn’t.

It is an expensive mistake, because the sign, together with the zeros, closes off whole regions of the plane. Done well, the rest of the function study (limits, asymptotes, derivatives) becomes almost a consistency check: if the graph you are drawing enters a zone forbidden by the sign, you already know you have gone wrong somewhere. First, though, you need to know how to get there, and where from: from the previous chapter, part 1: domain and symmetries, which is the absolute prerequisite for what follows: you can’t study the sign outside the domain.

Intercept with the y-axis

We begin with the simplest part, the one that is often dismissed in a line. The intercept with the y-axis is the value $f(0)$, that is the point $(0,f(0))$, but only if 0 belongs to the domain. If the domain excludes zero, the intercept with the y-axis simply doesn’t exist, and that must be written out, not left implied.

Example:

The domain is $x>0$ (because of the logarithm, which is more restrictive than $x\ne 0$). Zero doesn’t belong to the domain: no intercept with the y-axis, full stop. At this point many students write “it meets the y-axis at $x=0$” anyway, out of habit, without checking. It is a trivial mistake but a frequent one.

One detail worth fixing firmly in your mind, because it comes in handy later when you draw the graph: the intercept with the y-axis, if it exists, is only one. This follows directly from the definition of a function: each $x$ in the domain has only one value of $y$, so the vertical line $x=0$ can cross the graph in at most one point. With the x-axis there is no such constraint: the intercepts can be zero, one, two, or even infinitely many (think of $\sin x$).

Intercepts with the x-axis (the zeros)

Here the task is to solve $f(x)=0$, always inside the domain found in part 1. The technique depends on the form of the function.

Rational function. A fraction is zero when the numerator is zero, provided the denominator at that point is not zero as well. The classic mistake, which I see practically every week, is to set “the whole expression” to zero without distinguishing top and bottom, or to forget to check that the denominator doesn’t vanish at the same point (in which case you wouldn’t have a zero, but a point excluded from the domain). The correct procedure: solve numerator $=0$, then check that each solution you found makes the denominator non-zero.

Product of factors. The zero-product property applies: a product is zero if and only if at least one of the factors is zero:

It looks obvious written like this, but with products of three or four factors (perhaps with roots or exponentials inside as well) it is easy to lose one along the way.

Root, logarithm, exponential. Even students who are solid on the rest often slip here:

• $e^{g(x)}$ is never zero, for any value of $g(x)$: the exponential is a strictly positive function on its whole domain. If you find yourself “solving” $e^{g(x)}=0$, the answer is: no solutions, always. There is no value that makes an exponential vanish.
• $\ln (g(x))=0⟺g(x)=1$ (and not $g(x)=0$, which is instead the condition on the domain). Confusing these two conditions ($g(x)>0$ for the domain, $g(x)=1$ for the zero) is one of the most common mistakes in this chapter.
• A root $\sqrt{g(x)}=0⟺g(x)=0$, always checking that this value lies in the domain (where you need $g(x)\ge 0$).

Even or odd multiplicity. If a zero comes from a factor raised to a power, like $(x-2{)}^2$ or $(x-2{)}^3$, the multiplicity of that zero changes the behaviour of the graph right at that point:

• odd multiplicity (1, 3, 5…): the graph crosses the x-axis, and the sign changes from one side of the zero to the other;
• even multiplicity (2, 4…): the graph touches the x-axis and turns back (tangency), and the sign stays the same on both sides of the zero.

This links directly to the next section: multiplicity is already, in effect, information about the sign, not just about the shape of the graph near the zero.

When the equation can’t be solved by elementary methods. It happens, especially with mixed functions (for example $f(x)=x-\cos x$): there is no algebraic method that isolates $x$. In these cases the honest approach is not to “give up”, but to state explicitly that the zero exists thanks to the intermediate value theorem for zeros (Bolzano’s theorem): if $f$ is continuous on $[a,b]$ and $f(a)$ and $f(b)$ have opposite signs, then there is at least one zero in $(a,b)$. You find two values where the function changes sign, declare “there exists a zero $x_0\in (a,b)$ that cannot be expressed in closed form”, and, if you need an estimate, narrow the interval by bisection. It is not a loophole: it is the correct way to treat a case with no elementary solution, and an examiner will appreciate it more than an attempt at impossible algebra.

Sign study: the method

Once you have found the zeros and the points excluded from the domain, the task is to solve $f(x)>0$ (the case $f(x)<0$ is the complement). The standard method is the sign table, and it is worth doing it properly, because it is the tool you will use identically for every function from here on.

How it is built: factorise $f(x)$ into its factors (numerator, denominator, any separate factors of a product). Each factor becomes a row of the table. The columns are the intervals into which the x-axis is divided by all the “critical” values: the zeros of each factor and the points excluded from the domain. This is the point almost everyone forgets: the points excluded from the domain must be marked in the table exactly like the zeros, otherwise you lose the information about where the function changes behaviour because of a jump or an asymptote.

For each row, you mark with $+$ or $-$ the sign of the factor in each interval (one test value per interval is enough), and in the last row you apply the product/quotient rule for signs: an even number of negative factors gives a positive sign, an odd number gives a negative sign, exactly as in multiplying signed numbers. At the points excluded from the domain the final row stays “empty” (function not defined there), not zero.

Generic scheme (the symbols are only indicative):

Here $a$ is a zero and $b$ is a point excluded from the domain: in the column of $b$ the function is not defined; it isn’t zero.

Typical cases, worked out

1. Rational function

Domain: $x\ne -1$. Zeros: numerator $x^2-4=0⟺x=\pm 2$, both different from $-1$, so valid. Factors: $x^2-4=(x-2)(x+2)$ positive for $x<-2$ or $x>2$, negative between $-2$ and $2$; $x+1$ positive for $x>-1$, negative for $x<-1$.

So

zeros (multiplicity 1, hence crossing) at $x=\pm 2$.

2. With a logarithm

Domain: we need $x^2-3x+2>0$. Factorising, $(x-1)(x-2)>0⟺x<1$ or $x>2$.

Zeros: I solve $\ln (g(x))=0⟺g(x)=1$, that is $x^2-3x+2=1⟺x^2-3x+1=0$, which gives

Numerically $x_1\approx 0.38$ and $x_2\approx 2.62$; both fall in the domain (respectively $x<1$ and $x>2$), so they are valid zeros.

The key point, and the reason this is a good example: since $\ln$ is strictly increasing, the sign of $f$ comes down to comparing the argument with $1$, with no more need to think about the logarithm:

Intersecting with the domain ($x<1$ or $x>2$):

3. With an exponential

Domain: all of $\mathbb{R}$. Zeros: $(x-2)e^x=0$. But $e^x$ is never zero (as said above), so the only zero is $x=2$.

For the sign: $e^x>0$ for every $x$, so the exponential factor never contributes to a change of sign. You can “cancel” it from the reasoning and concentrate only on the other factor:

So $f(x)>0$ for $x>2$, $f(x)<0$ for $x<2$, zero (crossing) at $x=2$.

Tricks that save time

A few legitimate shortcuts, which I always use when marking:

• Factors that are always positive drop out of the sign reasoning, not of the computation of the zeros. Squares like $(x-1{)}^2$, exponentials $e^{g(x)}$, even roots $\sqrt{g(x)}$ (where defined): if a factor is always $\ge 0$ or always $>0$, it doesn’t affect the overall sign. You remove it from the table and work only on the factors that can really change sign.
• The sign of $-g(x)$ is the opposite of that of $g(x)$. It seems obvious, but on long expressions it helps to rewrite so as to take a minus sign out and reduce the number of cases.
• Even/odd symmetry carries over to the sign. If you have already established in part 1 that the function is even, the sign for $x<0$ is identical to that for $x>0$ (you only need to study half the domain and “mirror”); if it is odd, the sign flips when going from $x>0$ to $x<0$.

How to use the sign on the graph

This is the point I want you to take home more than any other: sign and zeros, taken together, already tell you in which half-plane each branch of the graph lives, before you have computed a single derivative.

In practice: when you draw the axes, shade (even just mentally, or with a light pencil) the zones where the function cannot be: above the x-axis on the intervals where the sign is negative, below the x-axis where it is positive. What remains is a corridor, often a narrow one, that the graph is forced to pass through. When you later draw the final graph with asymptotes and derivatives, that corridor is an automatic check: if the branch you are tracing leaves it, you have made a mistake upstream, not in the drawing.

Sign of f vs sign of f′ (a preview)

A clarification I always make explicitly, because it is the number one mistake of the whole series: the table of the sign of $f$ and the table of the sign of the derivative $f^{\prime }$ are two different tables, answering different questions. The first tells you where the graph lies above or below the x-axis. The second (which we will see later in the series) tells you where the function increases or decreases.

There is no contradiction in a function that is negative and increasing at the same time. Think of a branch that rises from $-10$ to $-1$: it is always below the x-axis (negative sign) while it increases (positive derivative). Keeping these two studies, and the two tables, separate avoids most of the confusion when you come to put all the information together in the final graph.

The mistakes I see most often

1. Setting the whole fraction to zero instead of just the numerator. $\frac{x-1}{x+2}=0$ does not have as solutions the zeros of the whole expression together: only $x=1$ (numerator), checking that $x=1$ does not make the denominator zero.
2. “Solving” $e^{g(x)}=0$. It has no solution, ever. If you find yourself looking for one, you set up the previous step wrongly.
3. Forgetting the points excluded from the domain in the sign table. Zeros alone are not enough: a point where the function is not defined still splits an interval and must be marked, otherwise the table “lies” about a whole stretch.
4. Confusing the domain condition with the zero condition in the logarithm. $g(x)>0$ is the domain, $g(x)=1$ is the zero: they are two different equations on the same argument, and they must be kept apart.
5. Using the sign table of $f^{\prime }$ to answer a question about the sign of $f$ (or vice versa). They are two independent studies; mixing them produces graphs that look plausible but are mathematically wrong.

How we work on this in lessons

• First the table done by hand on paper, row by row, before any shortcut: it is the only way to really absorb why the points excluded from the domain have to be there.
• We train the recognition of the cases (rational, product, root, logarithm, exponential) on mixed exercises, not one at a time, because in the test they arrive mixed up.
• We always connect sign and graph: after each table, I ask you to sketch freehand “where the graph can and can’t be” before moving on.
• Usually 1-2 lessons are enough to master this step on its own; the real leap in quality comes when it is combined with derivatives and asymptotes in the complete graph, which is how I approach final-year calculus in the maths programme.

The previous chapter is part 1: domain and symmetries; the next is part 3: limits and asymptotes. If function study is your weak point, get in touch for a free 30-minute introductory call.

The other parts of the series, if you need them out of order: part 4: monotonicity, maxima and minima, part 5: concavity and inflection points, part 6: the graph and the worked examples. To practise the sign on a real test, there is the Year 5 test on function study. Derivatives, which come in from part 4, have a guide of their own: derivatives explained properly. Limits and asymptotes, where the graph starts, seen from the test’s point of view, are in limits, continuity and asymptotes: the Year 5 test. If the sign of a fraction is your weak spot, the general case, with linear factors, is in rational inequalities in Year 3.