The topic of my Math Camp 2025 workshop is functions. Below is a list of research topic suggestions related to functions. The list is divided in half based on topics that have more of a "geometric" vs. "algebraic" flavour.

Contents

1. Geometry
   • Implicit equations
   • Tangent lines
   • Polar coordinates
   • Parametrised equations
   • Symmetries
2. Algebra
   • Polynomials
   • Integer sequences
   • Real sequences
   • Binary operations
   • General functions

1. Geometry

Implicit equations

As you know, the graph of a function $f\colon \mathbb{R}\to \mathbb{R}$ consists of the points $(x,y)$ in $\mathbb{R}^2$ such that $y=f(x)$.

There is a more general way of graphing functions. Take any function $F\colon\mathbb{R}^2\to\mathbb{R}$. Then, the equation $F(x,y)=0$ is called an implicit equation and one can graph the points $(x,y)$ which solve this equation.

Note that taking $F(x,y)=y-f(x)$, one can draw the graph of $f$ as the graph of an implicit equation.

1. Explore graphs of implicit equations using Desmos
   • You can get interesting phenomena even with just using 2-variable polynomials for $F$
   • One fun thing you can do with Desmos is you can type in an expression like $ax+by=0$ and create two sliders that let you change the values of $a$ and $b$
2. Find interesting plane curves online and study their graphs
   • Wikipedia has a list of curves
   • Implicit curves defined by degree 2 polynomials are the conic sections
   • In degree 3 there starts to be more geometrically interesting behaviour such as the "singularity" at the origin on the graph of $y^2-x^3-x^2=0$
   • Fun fact: one could even say that understanding the geometry of degree 3 curves played a major role in solving Fermat's Last Theorem

Tangent lines

Given a function $f\colon\mathbb{R}\to\mathbb{R}$ and a fixed real number $x_0$, it is interesting to try to find the tangent line to the graph of $f$ at the point $(x_0,f(x_0))$. The tangent line is a line that touches the graph at $(x_0,f(x_0))$ and is "pointed in the same direction" as the graph of $f$ at that point.

When $f$ is a polynomial function, there is an easy procedure for computing the tangent line at $(x_0,f(x_0))$. Firstly, the equation of a line through $(x_0,f(x_0))$ is given by $$(y-f(x_0))=m(x-x_0),$$ where $m$ is the slope of the line. Thus, we just need to decide what $m$ should be. Here is how you do it: Remove the constant term in $f$ and replace every monomial $x^k$ by $kx^{k-1}$ to obtain a new polynomial $Df$, called the derivative of $f$. Then, let $m$ be the number you get by plugging $x_0$ into the polynomial $Df$.

1. Try finding the tangent line to the graph of some chosen polynomial function at some chosen point using Desmos
   • Here is something harder but more informative: define a generic parabola in Desmos by typing $y=ax^2+bx+c$ and create sliders for $a$, $b$, and $c$. Then, draw the tangent to the parabola at $(0,c)$. See how changing the sliders changes the tangent line. What patterns do you spot? Ask for help if you get stuck at some point.
   • If some other student is reading about polynomial long division, you can make your job easier by collaborating with them and finding the tangent line by asking them to do polynomial long division as in this post for you
2. Choose an interesting 2-variable polynomial and a point on the graph of the implicit equation defined by the polynomial, then find the tangent line to the graph at the point
   • This is not that straight-forward, and I haven't found accessible explanations online explaining how to do this, so the only option you have is to ask for someone to help you do this
3. (Suggested by Niklas) As you learnt in school, for any given 2nd degree polynomial there is a formula for finding its roots. In fact, there are similar—although more complicated—formulas for the roots of 3rd and 4th degree polynomials. But after mathematicians had been trying for almost 300 years to find a similar formula for the roots of 5th degree polynomials, it was proven that such a formula cannot exist altogether. Therefore, since finding exact solutions to higher-degree polynomial equations is not possible in general, the best we can do is find approximate solutions. Tangent lines to functions provide a way of finding approximate solutions via Newton's method. See this LibreTexts article for details. This topic can be challenging, so you are encouraged to ask questions.

Polar coordinates

Recall that a function $f\colon\mathbb{R}\to\mathbb{R}$ can be graphed by drawing the point $(x,f(x))$ in $\mathbb{R}^2$ for each value of $x$. This depends on what we understand an expression $(x,y)$ to represent. As you learned in school $(x,y)$ means the point which is $x$ units away from the $y$-axis and $y$ units away from the $x$-axis. What we could do instead is say that $(\theta,r)$ represents the point which has distance $r$ from the origin and the angle between the line segment connecting the point to the origin and the non-negative $x$-axis is $\theta$. We can try graphing $f$ using this convention (i.e. looking at the point $(\theta,f(\theta))$) to get different looking plots.

1. Explore graphs in polar coordinates using Desmos
   • Try for example typing $r=\sin(a\theta)$ for different values of $a$ and the function will be plotted in polar coordinates
   • You can type $\theta$ in the expression by simply typing: theta

Parametric equations

A parametrisation is a function of type $\mathbb{R}\to\mathbb{R}^2$. The graph of a function $f\colon\mathbb{R}\to\mathbb{R}$ can be "parametrised" by declaring the parametrisation to be $r\colon\mathbb{R}\to\mathbb{R}^2$ defined by $r(t)=(t,f(t))$. You should convince yourself that if you draw for each real number $t$ a point at $r(t)$, then the resulting figure is precisely the graph of $f$. Of course, parametrisations allow for more general graphs than just the graphs of the functions $\mathbb{R}\to\mathbb{R}$.

1. Explore the graphs of parametrisations using Desmos
   • Say you want to graph the parametrisation $r(t)=(\sin t,\cos t)$. Then, you would write $(\sin t,\cos t)$ in Desmos. You will see that Desmos cannot actually graph the values of the parametrisation at every real number, but only on some interval, which you need to specify.
2. Find interesting parametric curves online
   • See for example the list in Wikipedia

Symmetries

Symmetry is a very general phenomenon that appears in different forms in various places. Taking symmetry into account is also a surprisingly powerful method in solving challenging mathematical or physical problems. Let us consider the familiar example of the symmetries of a regular $n$-sided polygon. One can imagine that the polygon consists of points in the plane $\mathbb{R}^2$. Then, its symmetries are rotations or reflections that move the points of the polygon back into itself. More specifically, a symmetry of a regular $n$-gon is a function $u\colon\mathbb{R}^2\to\mathbb{R}^2$ that either rotates every point about some fixed point or it reflects every point across some line. Furthermore $u$ must satisfy the requirement that if $p$ is a point on the $n$-gon, then $u(p)$ should also be on the $n$-gon. Given two symmetries $u_1,u_2\colon\mathbb{R}^2\to\mathbb{R}^2$, one can combine them to form a new symmetry by applying the first symmetry and the second symmetry right after. We can denote the obtained symmetry by $u_1u_2$.

For example take an equilateral triangle. One can rotate it by $120^\circ$, and we can denote this symmetry by $r\colon\mathbb{R}^2\to\mathbb{R}^2$. Next, one can reflect the triangle along the vertical line of symmetry and call the symmetry $s\colon\mathbb{R}^2\to\mathbb{R}^2$. It turns out that every other symmetry can be obtained by combining $r$ and $s$ with each other. For example, a $240^\circ$ rotation is given by $r^2$. The other two reflections are given by $sr$ and $sr^2$. Finally, denote by $e\colon\mathbb{R}^2\to\mathbb{R}^2$ the "identity function" which is defined by $e(x,y)=(x,y)$. These can be studied further by filling out the "multiplication table" for the symmetries $$\begin{array}{|c|c|c|c|c|}\hline \cdot&e&r&r^2&s&sr&sr^2\\\hline e\\\hline r\\\hline r^2\\\hline s\\\hline sr\\\hline sr^2\\\hline \end{array}$$

1. Fill out the multiplication table above
2. Take some other regular $n$-gon and list its symmetries. Let $r$ be a rotation that is a symmetry of the $n$-gon and $s$ be a reflection of the $n$-gon. Can you write all the other symmetries in terms of $r$ and $s$?
3. Come up with other symmetries
4. Check out the this interactive course on symmetries

2. Algebra

Polynomials

A polynomial $p\colon\mathbb{R}\to\mathbb{R}$ is a function of the form $$p(x)=a_0+a_1x+\cdots+a_nx^n$$ for some real numbers $a_0,\ldots,a_n$. Polynomials are interesting algebraically, because they can be added, subtracted and multiplied together. However, dividing one polynomial by another does not always result in a third polynomial. This is analogous to how division of integers does not always result in an integer. The fix is to consider division with remainder. The same can be done with polynomials.

Given a polynomial, it is interesting to ask what are its roots, i.e. the solutions in $x$ to the equation $p(x)=0$. This question is related to the problem of factoring polynomials. There is an important theoretical result regarding roots and factorisation, called the fundamental theorem of algebra, which states that every polynomial has a root, if you are willing to expand the definition of "numbers".

1. Learn about long division of polynomials
   • See Maths Is Fun and the example in the Wikipedia article
2. Figure out how to do modular arithmetic with polynomials
   • I haven't found any accessible source for this on the internet, so you would need to play around with polynomials and try to see if you can figure it out yourself
   • You can of course ask me or anyone else to help
3. Learn a new factoring method you haven't seen before
   • Brilliant
   • Paul's Online Notes
   • Rational root test on AoPS
   • Pascal's triangle and the Binomial Theorem
4. Learn about the complex numbers and fundamental theorem of algebra
   • Maths Is Fun
   • Paul's Online Notes

Integer sequences

An integer sequence can be thought of as a function of type $\mathbb{N}\to\mathbb{Z}$. There are two general ways of defining integer sequences. One can define a sequence by giving an explicit formula, for example the expression $$a_n=n^2$$ would define the sequence of square numbers. Another way would be to define a sequence via a recurrence relation. To do this, one specifies the first value(s) of the sequence and then gives a formula for $b_n$ in terms of the previous values in the sequence. For example one could define $b_0=1$, $b_1=1$ and $b_n=b_{n-1}+b_{n-2}$ for $n\geq 2$. This defines the famous Fibonacci sequence. It is a challenging and interesting problem to ask if a given recurrence relation can be given by an explicit formula.

1. Find an interesting integer sequence online. What properties does it have?
   • See the On-line Encyclopedia of Integer Sequences
2. Come up with your own sequence, compute few terms and try to spot patterns
3. Learn to solve simple linear recurrence relations
   • Note: this is an advanced problem
   • LibreTexts
   • AoPS
   • Wikipedia

Real sequences

Real sequences can be thought of as functions of type $\mathbb{N}\to\mathbb{R}$. Consider for example the following sequences. $$a_n=n^2,\quad b_n=(-1)^n,\quad c_n=\frac1{n+1},\quad d_n=(-2)^{-n}$$ The first sequences grows indefinitely while the second alternates between $1$ and $-1$. On the other hand, the sequences $c$ and $d$ begin as follows $$c\colon\quad1,\frac12,\frac13,\frac14,\frac15,\ldots$$ $$d\colon\quad 1,-\frac12,\frac14,-\frac18,\ldots$$ You may be able to notice that these sequences "approach" $0$ in some sense, while $0$ does not actually appear in the sequence at any point. This phenomenon is called convergence. $0$ is the limit of these sequences and this can be denoted as follows. $$\lim_{n\to\infty}c_n=\lim_{n\to\infty}d_n=0$$ The sequences $a$ and $b$ do not converge.

1. Write down sequences that do not converge and sequences that converge to a specific value, e.g. a sequence with limit $1$
2. Read about convergence and limits online
   • Limit of ratios of Fibonacci numbers
   • Paul's Online Notes (Calc I)
     • This discusses limits of functions in general, so you have to apply it in the special case of sequences
   • These lecture notes
     • I think it's good to skip page 1 first and see the examples and exercises on page 2
   • Paul's Online Notes (Calc II)

Binary operations

Addition, subtraction and multiplication can be thought of as functions of the form $X^2\to X$, where $X$ is any of $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$. More specifically, one could identify addition with a function $$A\colon X^2\to X$$ such that $A(x,y)=x+y$ and similarly for subtraction and multiplication. In general, any function of the form $X^2\to X$ is called a binary operation. Given a binary operation $B\colon X^2\to X$ it is common to use a symbol, e.g. $\oplus$, to write $B(x,y)=x\oplus y$.

When studying binary operations, one is interested in asking what kinds of properties they have. Here are some commonly studied properties:

Associativity. A binary operation $\oplus$ is associative, if for every elements $a$, $b$ and $c$ of $X$ we have $$a\oplus(b\oplus c)=(a\oplus b)\oplus c.$$ Commutativity. A binary operation $\oplus$ is commutative, if for every $a$ and $b$, $$a\oplus b=b\oplus a.$$ Identity. A binary operation $\oplus$ is said to have an identity element, if there is some $e$ in $X$ such that for every $a$ in $X$, we have $$a\oplus e=e\oplus a =a.$$ Inverses. A binary operation $\oplus$ with identity $e$ is said to have inverses, if for every element $a$ in $X$, there is $b$ such that $$a\oplus b=b\oplus a = e.$$

1. Which of the four properties do addition, subtraction and multiplication have?
2. Come up with a new binary operation and think about what properties it has
3. Try to come up with binary operations with different combinations of properties, i.e. one with identity but no inverses

General functions

Functions between sets have general constructions and properties that work for all functions. General functions can for example have the properties of being injective, surjective, or bijective. A function $f\colon X\to Y$ is injective, if one cannot find elements $x_1$ and $x_2$ in $X$ such that $x_1\neq x_2$ but $f(x_1)=f(x_2)$. $f$ is said to be surjective, if for every element $y$ of $Y$, there is some $x$ in $X$ such that $f(x)=y$. Finally, $f$ is bijective, if it is both injective and bijective. The meaning of injectivity and surjectivity is likely easier to be understood from an illustration like the one in this Wikipedia article. Bijectivity is extremely important, because for every bijection $f\colon X\to Y$, there is a function $g\colon Y\to X$, called the inverse of $f$, such that for any $x$ in $X$, $g(f(x))=x$ and for every $y$ in $Y$, $f(g(y))=y$. For example, the function $\sqrt{}\colon[0,+\infty]\to[0,+\infty]$ is the inverse of $(\cdot)^2\colon[0,+\infty]\to[0,+\infty]$.

An important construction of general functions is composition. Suppose $f\colon X\to Y$ and $g\colon Y\to Z$ are functions. Then, one can form the composition $g\circ f\colon X\to Z$ by defining $$(g\circ f)(x)=g(f(x)).$$ This is expressed diagrammatically as follows. For example one could take $X=\mathbb{N}$, $Y=\mathbb{Q}$, $Z=\mathbb{R}$ and $f(x)=\frac1x$ and $g(y)=\sqrt{y}$, and the composition will in this case be $$(g\circ f)(x)=\sqrt{1/x}.$$

To be able to study functions more effectively, it is useful to have some set theoretical notation available. Here are some useful notations If $x$ is an element of $X$, one writes $x\in X$. If $X$ and $Y$ are two sets such that every element of $X$ is an element of $Y$, then $X$ is said to be a subset of $Y$ and one writes $X\subseteq Y$. If $X$ is a subset of $Y$ but $X\neq Y$, one can write $X\subsetneq Y$. With these notations, we can define images and pre-images. If $h\colon X\to Y$ is a function and $U\subseteq X$ is a subset, then the image of $U$ under $h$ is the subset of $Y$ consisting the elements $y\in Y$ such that there is $u\in U$ satisfying $h(u)=y$. The image is denoted by $h(U)$. For example, the image of $\mathbb{R}$ under the map $s\colon \mathbb{R}\to\mathbb{R}$ given by $s(x)=x^2$ is the subset of non-negative real numbers: $$s(\mathbb{R})=\left\{\text{non-negative real numbers}\right\}.$$ Next, if $V\subseteq Y$ is a subset of $Y$, then its pre-image under $h$ is the subset of $X$ consisting all $x\in X$ such that $h(x)\in V$. The pre-image is denoted by $f^{-1}(V)$. For example, the pre-image of the interval closed $[0,2]$ under the map $s$ is $$s^{-1}([0,2])=[-\sqrt{2},\sqrt{2}].$$

1. Take your favourite functions and check if they are injective, surjective, bijective, or neither
2. Take your favourite functions, choose subsets of domains and codomains, and figure out what their images or pre-images are
3. Prove that if $f\colon X\to Y$ and $g\colon Y\to Z$ are both injective, then so is their composition $g\circ f\colon X\to Z$. Show that the same holds if "injective" is replaced by surjective or bijective
4. Prove that if $f\colon X\to Y$ and $g\colon Y\to Z$ are functions such that $g\circ f$ is injective, then $f$ is injective. Similarly, if $g\circ f$ is surjective, then $g$ is surjective
5. Prove that if $h\colon X\to Y$ is a function and $U\subseteq X$ is a subset, then $U\subseteq f^{-1}(f(U))$. Find an example, where $U\neq f^{-1}(f(U))$
6. Prove that if $h\colon X\to Y$ is a function and $V\subseteq Y$ is a subset, then $f(f^{-1}(V))=V$
7. Read about how bijections between infinite sets lets us compare the sizes of infinities
   • Mathigon
   • Maths Is Fun