2.1 Functions and their graphs

A function $f$ is a rule of correspondence that associates with each element $x$ in some set, called the domain, a single value $f(x)$ from a second set, called the codomain. In the following, we consider functions of the form $f:I\to ℝ$, where the domain $I$ of $f$ is a subset of $ℝ$.

In general, how do we go about sketching the graph of a function $f(x)$ by hand? First of all, we look for any obvious roots of $f(x)=0$: we might calculate the values of $f(0)$ and $f(\pm 1)$, say. This is just to get some idea for the position of the graph in relation to the axes. In order to get an idea of the shape of the curve, we look for turning points.

Turning points

Turning points are special types of stationary points:

Important point: A necessary part of the definition of a stationary point is that $f^{\prime }(\alpha )$ be defined, i.e. that $f$ be differentiable at $\alpha$.

For $\delta$ small (positive or negative), ${\delta }^2(\delta +2)\ge 0$. Thus $f(x)\ge f(1)$ for $x$ sufficiently close to 1. Hence $f(1)=0$ is a local minimum of $f(x)$ at $1$. On the other hand, $f(x)\to -\mathrm{\infty }$ as $x\to -\mathrm{\infty }$, so $0$ is certainly not a global minimum.

A stationary point may be a local maximum, a local minimum or a saddle point (which is neither local minimum nor local maximum). A stationary point which is a local maximum or a local minimum is called a turning point. There are a few different ways to determine which type of stationary point $(\alpha ,f(\alpha ))$ is.

Let’s sketch some curves to see the different things that might happen.

The above graphs are a visual justification of the rule you probably already know for determining whether a point is a local minimum, a local maximum or a point of inflection:

Case 1. If then $(\alpha ,f(\alpha ))$ is a local maximum;

Case 2. If $f^{\mathrm{\prime \prime }}(\alpha )>0$ then $(\alpha ,f(\alpha ))$ is a local minimum;

Case 3. If $f^{\mathrm{\prime \prime }}(\alpha )=0$ then $(\alpha ,f(\alpha ))$ may be a local maximum or minimum, or it could be a saddle point. In this case we need to look at values of $f(x)$ for $x$ just greater than or just less than $\alpha$ to determine what type of point $(\alpha ,f(\alpha ))$ is.

We now calculate the values of $f(-1)=-1+3-1=1$ and $f(1)=1-3-1=-3$. Let’s plot these points on the graph, bearing in mind that $f(x)\to -\mathrm{\infty }$ as $x\to -\mathrm{\infty }$ and $f(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$. We may as well calculate $f(0)$ too.

We can see that $x=-1$ must be a local maximum and $x=1$ must be a local minimum. Finally, we may wish to calculate some further values $f(2)$, $f(-2)$ etc. in order to estimate where the curve crosses the $x$-axis, i.e. where the roots of $f(x)=0$ are. We have $f(2)=1$ and $f(-2)=-3$, so there is (at least) one root of $f(x)=0$ for $x\in (-2,-1)$, there is at least one root for $x\in (-1,0)$ and at least one more root for $x\in (1,2)$. Since a cubic polynomial can have at most three real roots, we see that $f(x)=0$ has exactly one root in each of the intervals $(-2,-1)$, $(-1,0)$ and $(1,2)$.

The above discussion gives us a general approach to sketching the graph of a function:

- first calculate some values of $f(x)$ at key points,

- then look for turning points,

- finally, determine what happens as $x\to \pm \mathrm{\infty }$.

A note of caution: All of the above discussion assumes that $f(x)$ is differentiable. If it isn’t, then there may be local (or even global) extrema for which $f^{\prime }(x)$ doesn’t exist.