MAT1001 Differential Calculus: Lecture Notes

Chapter 2 Functions

2.1 What is a function

The concept of a function is essential to understand not just calculus but also computer programming. We can think of a function as being a black box that takes in information, potentially changes it in some way, and then outputs information.

Being more precise, a function is a map between two spaces, the domain, and the codomain. e.g.

\begin{equation*} f:X\to Y \end{equation*}

and it sends a point \(x\in X\) to a point \(f(x)=y\in Y\). Functions have the property that they map a point \(x\in X\) to a single point \(y\in Y\),for example:

\begin{equation*} y=f(x)=x^{2}+1, \end{equation*}

is a function since every value of \(x\) gives a single value of \(y\). However,

\begin{equation*} y^{2}=x+1, \end{equation*}

is not a function since every \(x\) corresponds to two values of \(y\), e.g. if \(x=3\) then \(y^{2}=3+1=4\) and \(y\) can be both \(2\) and \(-2\).

Note that when we write \(f(x)\) we are not saying \(f\) times \(x\), but mean that \(f\) is a function of \(x\), e.g. \(f\) takes in a value of \(x\) and returns a number \(y=f(x)\).

Throughout this module we will meet several different types of functions so you will need to get familiar with this notation and understand how to evaluate functions. There will be some questions in the tutorial sheet to help you practice this.

A key property of a function is its Roots or zeros, these are the values of \(x\) such that \(f(x)=0\). For linear functions finding the roots is just a matter of rearranging the equation, sometimes referred to as changing the subject. In this module I am assuming that you have some familiarity with this, if not take a look at the background material in chapter 8 or ask me to point you towards more resources. There will also be some revision questions on this topic in the first couple of weeks tutorials.

If you do not remember how to use the quadratic formula then I suggest that you look at the background material in chapter 8.

If this was a course for mathematicians this is where we would spend time talking about the domain and range of a function. How they are defined, and when we need to be careful to avoid dividing by zero. Here, we will not go through this, and I will just remind you that in your algebraic manipulations you should not divide by any quantity that is zero.

A simple, yet useful, class of functions are the rational functions. There are functions who are the ratio of two polynomials, e.g.

\begin{equation*} f(x)=\frac {4x+10}{x^{2}-2x-15}. \end{equation*}

A useful way to understand a function is draw a graph. In a graph, the domain of the function \(X\) is drawn horizontally and the codomain, \(Y\), is drawn vertically, using Cartesian axes \((x,y)\). The graph consists of the points \((x,y)\) with \(y=f(x)\). In many books \(x\) is called the independent variable and \(y\) the dependent variable. In the tutorials we will discuss how to produce plots using a computer, either using MATLAB, Python, or WolframAlpha. There are also online graphing programs like desmos and GeoGebra. In these notes most of the graphs are produced using the Tikz package for LaTeX, which is a typesetting and markup language¹.

Note that the equation \(y^{2}=x+1\) also defines a parabola, see fig. 2.3. However if we plot this function the curve is not the graph of a function. You should think about why this is.

The concept of an inverse function makes intuitive sense. However, if we pretend to be mathematicians and treat this carefully we quickly encounter some problems. For example, what happens if we have a function which sends two different values of \(x\) to the same value? Then we are unable to know which value we started with, so cannot build an inverse function. For example \(f(x)=x^{2}\) sends both \(x\) and \(-x\) to \(x^{2}\) so we cannot find an inverse that works everywhere, this is why when we firts meet the square root function you often see it as \(\pm \sqrt {\phantom {+}}\).

Mathematicians fix this by introducing the notion of a One-to-one function, sometimes called an injective function. A function is called one-to-one if no two values of \(x\) produce the same value of \(y\), so that

\begin{equation*} f(x_{1})\neq f(x_{2}) \qquad \text { for } x_{1}\neq x_{2}. \end{equation*}

The advantage of one-to-one functions is that we can find inverses for them. If we have two one-to-one functions which satisfy

\begin{equation*} \left (f\circ g\right )(x)=x=\left (g\circ f\right )(x), \end{equation*}

then \(f\) and \(g\) are inverses and we write \(g(x)=f^{-1}(x)\).

2.2 Trigonometric functions

Now that we have discussed functions and how to graph them, we can focus on some specific functions which it is very common to encounter.

From your previous maths experience you have probably come across Trigonometric (trig) functions in the context of triangles, where they are used to calculate lengths and angles. If you do not remember how this works then I suggest that you have a look at the Triganometry primer in chapter 8 or check out some revision material available here. Another important thing to remember is that you should always work in radians rather than degrees when using trig functions. This is because radians are a more natural unit for angles and if we did not use them lots of formulas would need extra factors of \(\uppi \) to be added for them to be valid.

In the setting of triangles the angles are restricted to run between \(0\) and \(\uppi \) radians as for angles greater than this we would no longer have a triangle. However, the functions are valid for any real values of the argument, \(x\). Typically we will be interested in angles within the range \([0,2\uppi )\), but need to remember that, as shown in fig. 2.6, these functions are \(2\uppi \) periodic. This means that

\begin{align*} \sin (x+2\uppi )=\sin (x),\\ \cos (x+2\uppi )=\cos (x),\\ \tan (x+2\uppi )=\tan (x). \end{align*}

We can think of these as functions from \([0,2\uppi )\to [-1,1]\) which reduce to the familar trig functions for angles between \(0\) and \(\uppi \).

The trig functions are clearly not one-to-one, since they are periodic, in fact even restricted to \(x\in [0,2\uppi )\) there are repeated values. However, just like with the square root, it is convenient to introduce inverse functions, where we need to be careful which quadrant our angle is in to get a single value out. These inverse functions are often denoted

\begin{align*} \sin ^{-1}(x)&=\arcsin (x),\\ \cos ^{-1}(x)&=\arccos (x),\\ \tan ^{-1}(x)&=\arctan (x). \end{align*}

Your calculator will have a button for each trig function, and a way to select the inverse functions, typically calculators return answers in the following ranges:

\begin{equation*} 0\leq \arccos (x)\leq \uppi ,\qquad -\frac {\uppi }{2}\leq \arcsin (x)\leq \frac {\uppi }{2}, \qquad -\frac {\uppi }{2}< \arctan (x)< \frac {\uppi }{2}. \end{equation*}

The inverse trig functions are useful if we need to solve equations involving trig functions.

You will have the opportunity to practice more problems like this either by going through the week one tutorial sheet or by looking at Section 1.5 in [Dawkins, 2025a].

The trig functions satisfy some nice relationships that you should try to learn, and which we quote here without any proof. Some of these are fairly easy to prove and are left as an exercise, while others are harder and can be looked up if you are curious.

The first one is that

\begin{equation*} \tan (x)=\frac {\sin (x)}{\cos (x)}. \end{equation*}

Squares: The squares of trig functions have the nice property that

\begin{equation*} \sin ^{2}x+\cos ^{2}x =1, \end{equation*}

which is a consequence of Pythagoras’ theorem.

Multiple angles: There are some very useful identities when we consider trig functions for the sum and difference of an angle:

\begin{align*} \sin \left (x\pm y\right )&=\sin x \cos y \pm \cos x \sin y,\\ \cos \left (x\pm y\right )&=\cos x \cos y \mp \sin x \sin y. \end{align*} For the special case of \(x=y\) this leads to

\begin{align*} \sin 2x &= 2\sin x \cos x,\\ \cos 2x &=\cos ^{2}x-\sin ^{2}x = 2\cos ^{2}x -1 =1-2\sin ^{2}x. \end{align*}

You should try to become familiar with these identities, at least for this module, as they will be useful wherever trig functions appear.

Further trig functions: There are three more trig functions that you should be aware of. Again, these are not independent trig functions but are one over the familiar trig functions. These are:

\begin{align*} \sec x &=\frac {1}{\cos x},\\ \csc x &=\frac {1}{\sin x},\\ \cot x &=\frac {1}{\tan x}. \end{align*} Plots of these three functions are shown in fig. 2.8.

2.3 Logarithms and Exponentials

Another pair of very useful functions which we need to be familiar with are the logarithm and the exponential function.

In your previous maths courses you may have come across the fact that we call powers exponents, e.g. in an expression \(f(x)=x^{b}\) then \(b\) is called the exponent„ or the index. An exponential function is then a function like

\begin{equation*} f(x)=b^{x}, \end{equation*}

where a constant⁴ \(b\) is raised to the power of the variable \(x\). In this case the number \(b\) is called the Base of the exponential function.

The exponential function with base \(2\) is shown in fig. 2.9. It turns out that by appropriate scaling of \(x\) different bases are related⁵ so we only need one standard basis.

When working with binary the basis \(b=2\) is chosen, while in scientific notation \(b=10\) is chosen. Nowadays the basis in general use is denoted \(e\) or \(\exp \) with \(\exp (x)\) called the exponential function.

The reason for picking \(e\) as the basis is that simplifies some of the algebra associated with exponential functions, it also has the advantage that the tangent to the curve has gradient \(1\) at \(x=0\). This last part boils down to

\begin{equation} e\simeq \left (1+h\right )^{\frac {1}{h}} \label {eq: exp approximation} \end{equation}

for \(h\) close to zero.

Closely related to \(\exp \) is its inverse function, called the logarithm function. You will see it written as either \(\log (x)\) or \(\ln (x)\), or rarely as \(\log _{e}(x)\) where the base is made explicit.

As this is an inverse function it is defined as the solution to the equation \(e^{y}=x\). For example \(e^{y}=4\) is solved by \(y=\ln (4)\).

The graphs of both \(\exp (x)\) and \(\ln (x)\) are shown in fig. 2.10. Note that we are, currently, not defining \(\ln \) when \(x\) is negative ⁶. You may have guessed that we could take a logarithm in a different basis since it is defined so that

\begin{equation*} \ln (\exp (x))=x=\exp (\ln (x)). \end{equation*}

This is why there are several slightly different ways of writing the logarithm. If you learnt about it in school you probably used \(\log (x)\) to mean log with base \(10\). In this module if we work with a basis other than \(e\) we will make this explicit by writing \(\log _{b}(x)\) to mean log with base \(b\).

To be clear \(\log _{b}(x)\) is the function defined such that

\begin{equation*} \log _{b}(b^{x})=x=b^{\log _{b}(x)}. \end{equation*}

A nice property of exponentials and logarithms is that they map addition to multiplication. What this means is that

\begin{align*} \exp (a)\exp (b)&=\exp (a+b),\\ \ln (ab)&=\ln (a)+\ln (b),\\ \ln \left (\frac {a}{b}\right )&=\ln (a)-\ln (b). \end{align*}

Some of the other properties of these functions are that:

\begin{align*} \exp (-x)&=\frac {1}{\exp (x)},\\ \exp (x)&>0,\\ \exp (x)&\to \infty \quad \text {as } x\to \infty ,\\ \exp (x)&\to 0 \quad \text {as } x\to -\infty ,\\ \ln (x) &\to \infty \quad \text {as } x\to \infty ,\\ \ln (x)&\to -\infty \quad \text {as } x\to 0^{+}. \end{align*} We will return to discussing limits shortly, but here the statement \(\exp (x)\to \infty \) as \(x\to \infty \) means that the function \(\exp (x)\) becomes infinitely large as \(x\) becomes infinitely large. The terminology for this is that \(\exp (x)\) diverges as \(x\) tends to infinity.

The final, useful, identity about logarithms is how to change the basis. The change of basis formula is

\begin{equation} \log _{b}(x)=\frac {\log _{a}(x)}{\log _{a}(b)}. \label {eq:log change of base} \end{equation}

As with trig functions in section 2.2, \(\ln \) and \(\exp \) are useful when solving equations. There are lots of examples in [Dawkins, 2025a] but we will go through a couple of examples here, there will be more on the tutorial sheets.

It may seem like example 2.7 took quite a bit of working out, but with practice you will get faster at solving problems like this. In the lectures you will frequently hear me paraphrasing George Pólya and saying that Mathematics is not a spectator sport. This means that while reading these notes and attending the lectures can help you in your learning there is no substitute for actually rolling your sleeves up and solving problems. Remember that most modules expect you to be putting in around three hours of self study for every hour of contact time.

2.4 Hyperbolic functions

The next class of functions that it is useful to know about are the hyperbolic trig functions. As shown in fig. 2.11, the hyperbolic trig functions are the analogue of the standard trig functions of section 2.2 but adapted to the geometry of the hyperbola, \(x^{2}-y^{2}=1\) rather than the circle \(x^{2}+y^{2}=1\). For the standard trig functions, their argument was an angle related to how far round the unit circle we had gone. In the hyperbolic case the angle is now twice the shaded area shown in section 2.2.

There is a hyperbolic equivalent of all of the standard trig functions, and the notation is very similar just with an added \(h\) at the end:

• • hyperbolic sine \(\sinh \),

• • hyperbolic cosine \(\cosh \),

• • hyperbolic tangent \(\tanh \),

• • inverse hyperbolic sine \(\arcsinh \),

• • inverse hyperbolic cosine \(\arccosh \),

• • inverse hyperbolic tangent \(\arctan \).

There are also hyperbolic versions of the other trig functions but we will not be as concerned with them here.

As in the regular trig case the hyperbolic tangent is defined in terms of the other two functions as

\begin{equation} \tanh (x)=\frac {\sinh (x)}{\cosh (x)}. \label {eq: hyperbolic tangent} \end{equation}

The hyperbolic trig functions are actually defined in terms of the odd and even parts of the exponential function⁸ as follows:

\begin{align} \sinh (x)&=\frac {e^{x}-e^{-x}}{2}, \label {eq: sinh exp}\\ \cosh (x)&=\frac {e^{x}+e^{-x}}{2}. \label {eq: cosh exp} \end{align} They have the following useful properties:

\begin{align*} \sinh (-x)&=-\sinh (x),\\ \cosh (-x)&=\cosh (x),\\ \cosh ^{2}(x)-\sinh ^{2}(x)&=1,\\ 1-\tanh ^{2}(x)&=\sech ^{2}(x). \end{align*} These are very similar to the identities satisfied by the ordinary trig functions. However, the sign in front of \(\sinh (x)\) and \(\tanh (x)\) is negative while that in front of \(\sin (x)\) and \(\tan (x)\) in the equivalent formulae was positive. The rule of thumb when converting from identities for trig functions to identities for hyperbolic trig functions is to send \(\cos ^{2}(x)\to \cosh ^{2}(x)\) and \(\sin ^{2}(x)\to -\sinh ^{2}(x)\). There are also analogues of the addition of angle identity formulas that you can try to derive if you are interested.

Since the hyperbolic trig functions are related to the exponential function you can probably guess that their inverses are related to the logarithm. It is worth thinking about how to show this relationship. I will show you how to do this for \(\sinh (x)\) and \(\tanh (x)\) but leave deriving the identify for \(\cosh (x)\) as an exercise for the interested reader.

If you do a similar calculation for \(y=\arccosh (x)\) you will find that

\begin{equation} \arccosh (x)=\ln \left (x\pm \sqrt {x^{2}-1}\right ). \label {eq: arccosh log} \end{equation}

You should think about why there is a \(\pm \) in this formula but there was not one in eq. (2.6).

You may be asking why we have spend the time deriving these identities for the hyperbolic functions, and why we discussed their relationship with the exponential function. This is because when we start to differentiate or integrate hyperbolic functions it is often more straightforward to use the expressions involving exponentials and logarithms.

2.5 Limits and asymptotics

We have now spent quite a bit of time discussing different examples of functions and their properties, as well as how to plot them. If we consider the plot of the tangent function in fig. 2.7 you may have asked at the time, what happens when the red line goes off the top of the page and reappear at the bottom? If we followed both lines we would see that they are getting closer and closer to the point \(x=\uppi /2\). This idea of zooming in on a particular value \(x=a\) and asking what happens to a function there is known as taking a limit as \(x\) approaches \(a\) and is denoted \(\lim _{x\to a}f(x)\).

If the function is well behaved at the point \(a\), then the limit is just the value of the function evaluated at \(a\), \(f(a)\). For example

\begin{equation*} \lim _{x\to 0}\cos (x)=1=\cos (0). \end{equation*}

The value of the limit does not have to be finite, it can head off to infinity. A good example of this is the exponential function which just keeps getting larger as \(x\) increases. We denote this by

\begin{equation*} \lim _{x\to \infty }\exp (x)\to \infty . \end{equation*}

Notice that since the value of the limit is infinite we do not write an equals sign but instead use \(\to \) to denote that the limit diverges⁹. There are lots of other examples of functions that diverge, and not always for large \(x\). A nice example to have in mind is \(y=1/x^{2}\) which diverges as \(x\to 0\).

In the case of \(\tan (x)\) we can observe that \(\lim _{x\to \uppi /2}\tan (x)\) will take on different values depending on if \(x\) is approaching \(\uppi /2\) from above or below. This means that the limit does not exist. In section 2.6 we will discuss the consequences of this for the function in more detail. If we are taking the limit from above we often say \(x\to a^{+}\), while the limit from below is denoted \(x\to a^{-}\). In this case

\begin{align*} \lim _{x\to \frac {\uppi }{2}^{+}}\tan (x)&\to \infty ,\\ \lim _{x\to \frac {\uppi }{2}^{-}}\tan (x)&\to -\infty , \end{align*} so \(\tan (x)\) diverges in different directions depending on how we approach \(\uppi /2\).

Curve sketching is a useful way to understand the main features of a function and can help us to know if the limit exists and if we have the same limit when approaching from different directions.

The general rule when curve sketching is to set \(y=f(x)\) for the function of interest \(f(x)\), and then to investigate the following:

• • The intercepts of \(f(x)\), these are the points where the function meets or crosses the coordinate axes, e.g. \(f(x=0)\) and \(f(x)=0\), as these position the graph relative to the coordinate axes.

• • What happens to \(y\) as \(x\to \pm \infty \)?

• • Which values of \(x\) make \(y\to \pm \infty \)?

• • Does the function have any symmetry? For example does \(f(-x)=f(x)\) (\(f\) is even), or \(f(-x)=-f(x)\) (\(f\) is odd).

• • Is the function periodic?

Note that a function can have several of these interesting features. e.g. the trig functions are all periodic, while \(\sin (x)\) and \(\tan (x)\) are odd functions and \(\cos (x)\) is an even function.

As a warning, this approach of tabulating the values does not work if the function is oscillating. For example, if you try to evaluate the limit as \(x\to 0\) of \(\cos \left (\uppi /x\right )\) it can look like it is tending to a constant if we are not careful with the values that we pick. However, if we plot the function we see that it is highly oscillatory around zero.

Remember that when evaluating a limit we want to look for as many cancellations as possible to simplify the calculation. This can be looking for common factors between the numerator and denominator, but can also mean fully expanding out all terms as there may be cancellations hidden by the way the function has been written.

Often it is useful to plot a function when we are estimating the limit, while this is not compulsory it can be very helpful, particularly if as in the case of \(\tan (x)\), the value of the limit depends on the direction of approach.

There are a few limits that we need to know that we will not prove here, time permitting we will prove these in chapter 7, but you can also look at the “Proof of Trig Limits” section of Dawkins [2025a] to see one approach to proving them. These limits are

\begin{align} \lim _{x\to 0}\frac {\sin x}{x}&=1, \label {eq: sin limit}\\ \lim _{x\to 0}\frac {\cos x -1}{x}&=0, \label {eq: cos limit}\\ \lim _{h\to 0}\frac {e^{h}-1}{h}&=1, \label {eq: exp limit}\\ \lim _{h\to 0}\frac {\ln (1+h)}{h}&=1. \label {eq: ln limit} \end{align} Note that eq. (2.13) is equivalent to eq. (2.1) which we discussed above as one of the definitions of the exponential function.

2.6 Continuity and differentiability

Now that we have some examples of functions and understand how to take limits, we can define two properties of a function which will be very important later in the course: continuity and differentiability.

A function \(f(x)\) is said to be continuous at a point \(x=a\) if

\begin{equation} \lim _{x\to a}f(x)=f(a). \label {eq: continuity} \end{equation}

If \(X\) is the domain of \(f(x)\), we say that \(f\) is continuous on \(X\) if it is continuous at each point in \(X\), often we would just say that \(f\) is a continuous function. As a rule of thumb, we can say that a function is continuous if its graph can be drawn from start to finish without taking your pen off the paper. Functions which are not continuous will have jumps or divergences at the point that fails to be continuous.

If we have a rational function then we can find where it is not continuous, called being discontinuous, by finding the roots of the denominator.

Related to continuity is differentiability. A function \(f(x)\) is differentiable at a point \(a\) if the limit

\begin{equation*} \lim _{x\to a}\frac {f(x)-f(a)}{x-a} \end{equation*}

exists. We call this limit the derivative of the function at the point \(a\),

\begin{equation} f'(a)=\lim _{x\to a}\frac {f(x)-f(a)}{x-a}, \label {eq: rate of change at a} \end{equation}

the notation \(\frac {\ud f}{\ud x}(a)\) is sometimes used instead of \(f'(a)\). A function is called continuously differentiable if its derivative is also continuous.

It is important to note that differentiability at \(a\) implies continuity at \(a\), but continuity does not imply differentiability. For example the absolute value function shown in fig. 2.14 is continuous everywhere, but is only differentiable everywhere except at \(x=0\).

The fraction in eq. (2.17) may look familiar to you. If we had a straight line \(y=f(x)=mx+c\) where \(m\) is the gradient of the straight line and \(c\) is the \(y\)-intercept, then calculating this fraction gives

\begin{equation*} \frac {f(x)-f(a)}{x-a}=\frac {mx+c-(ma+c)}{x-a}=\frac {m(x-a)}{x-a}=m, \end{equation*}

which is the gradient of the curve. For a function that is not a straight line, this procedure gives the gradient of the straight line between \(x\) and \(a\). In the limit that \(x\to a\) this fraction becomes the gradient of the tangent line¹³ to the curve at \(a\). See fig. 2.16 for the example of the tangent to parabola \(y=x^{2}\).