Representing a Function

Analytical representation of a function.

This is the most common way to represent a function. We define a formula that contains arithmetic operations on constant values and the variable $$${x}$$$ that we need to perform to obtain the value of $$${y}$$$.

For example, $$${y}=\frac{{1}}{{{1}+{{x}}^{{2}}}}$$$, $$${y}={3}{{x}}^{{2}}-{5}$$$, etc.

Now, from the function $$${y}=\frac{{1}}{{{1}+{{x}}^{{2}}}}$$$, we can find that $$${y}{\left({2}\right)}=\frac{{1}}{{{1}+{{2}}^{{2}}}}=\frac{{1}}{{5}}$$$.

However, this is not the only way to represent a function. We can represent it analytically without a formula.

For example, consider the function $$${y}={\left[{x}\right]}$$$ (floor function). Clearly, $$${\left[{1}\right]}={1}$$$, $$${\left[{2.5}\right]}={2}$$$, $$${\left[\sqrt{{{13}}}\right]}={3}$$$, $$${\left[-\pi\right]}=-{4}$$$, etc., but there is no formula that expresses $$${\left[{x}\right]}$$$.

Another example is $$${f{{\left({x}\right)}}}={x}!={1}\cdot{2}\cdot{3}\cdot\ldots\cdot{x}$$$. The domain of this function is a set of natural numbers, because $$${x}$$$ can take only natural values. So, $$${f{{\left({3}\right)}}}={3}!={6}$$$, but again, there is no formula that expresses $$${x}!$$$.

Now, let's talk about the domain of the function.

We can state the domain of the function explicitly, for example, $$${f{{\left({x}\right)}}}={{x}}^{{2}},{3}\le{x}\le{5}$$$. This means that $$${x}$$$ can change only in the interval $$${\left[{3},{5}\right]}$$$.

But often, the domain is given implicitly, i.e. the domain is all values of $$${x}$$$ where $$${f{{\left({x}\right)}}}$$$ is defined.

For example, for the function $$${f{{\left({x}\right)}}}={{x}}^{{2}}+{1}$$$, the domain is a set of all real numbers, i.e. the interval $$${\left(-\infty,\infty\right)}$$$, because the function is defined for any value of $$${x}$$$.

The domain of the function $$${h}{\left({x}\right)}=\frac{{1}}{{{x}-{2}}}$$$ is all real numbers except $$${x}={2}$$$, i.e. $$${\left(-\infty,{2}\right)}\cup{\left({2},\infty\right)}$$$. $$${x}={2}$$$ is not in the domain, because $$${h}{\left({2}\right)}$$$ is not defined (the denominator equals $$${0}$$$).

The domain of the function $$${y}{\left({x}\right)}=\sqrt{{{1}-{{x}}^{{2}}}}$$$ is $$${\left[-{1},{1}\right]}$$$, because for these values the expression under the root is non-negative, and, thus, the function $$${y}{\left({x}\right)}$$$ is defined.

Moreover, in a real-world application, we also need to make sure that the function makes sense. For example, consider the law of free fall $$${s}=\frac{{1}}{{2}}{{g{{t}}}}^{{2}}$$$. This function is defined for all $$${t}$$$ but it doesn't make sense when $$${t}<{0}$$$ (time can't be negative). Therefore, the domain of the function in the given situation is $$${\left({0},\infty\right)}$$$.

Let's work another quick example.

One more quick example for practice.

And let's do another short example before passing on.

Numerical (tabular) representation of a function.

In real life, the dependence between variables is formed based on experiments or observations. For example, at any given time $$${t}$$$, we can calculate the world population $$${N}$$$. We say that $$${N}$$$ is a function of $$${t}$$$ but can't find an explicit formula to express this. All we have is a table of values as below:

However, in further topics of calculus, we will be able to find a function that will approximate these values.

Graphical representation of a function.

Although in calculus functions are not defined graphically, graphical representation is very helpful in studying functions.

Suppose that we are given a function $$${y}={f{{\left({x}\right)}}}$$$ with the domain $$${X}$$$. Imagine a plane with two perpendicular axes: the x-axis and the y-axis. Consider a pair of corresponding values $$${x}$$$ and $$${y}$$$, where $$${x}$$$ is taken from set $$${X}$$$ and $$${y}={f{{\left({x}\right)}}}$$$. The image of this pair is the point $$${A}{\left({x},{y}\right)}$$$ with the x-coordinate $$${x}$$$ and the y-coordinate $$${y}$$$. When the variable $$${x}$$$ is changing within the interval $$${X}$$$, this point describes some curve. This curve is the graph of the function $$${y}={f{{\left({x}\right)}}}$$$.

So, to draw the graph of the function, take from the interval $$${X}$$$ points that are close to each other: $$${x}_{{1}},{x}_{{2}},{x}_{{3}},\ldots{x}_{{n}}$$$. Now, find the corresponding y-values: $$${y}_{{1}}={f{{\left({x}_{{1}}\right)}}}$$$, $$${y}_{{2}}={f{{\left({x}_{{2}}\right)}}}$$$,..., $$${y}_{{n}}={f{{\left({x}_{{n}}\right)}}}$$$.

Draw the points $$${\left({x}_{{1}},{y}_{{1}}\right)}$$$, $$${\left({x}_{{2}},{y}_{{2}}\right)}$$$,..., $$${\left({x}_{{n}},{y}_{{n}}\right)}$$$. Connect these points by a smooth curve. We've obtained the graph of the function. Note: the more points you take, the more accurate graph you will obtain.

Let's move on to practical examples.

And a bit more work to memorize it better.

The graph of the function is a curve in the xy-plane. But the question arises: which curves in the xy-plane are graphs of functions? This is answered by the following test.

Graphical representation allows us to determine easily whether the curve is a function.

Consider the graph of the function $$${{y}}^{{2}}+{{x}}^{{2}}={4}$$$. It is a cirlcle with a radius of 2. Clearly, it is not a function.

To see why it is so, express $$${y}$$$ in terms of $$${x}$$$: $$${{y}}^{{2}}={4}-{{x}}^{{2}}$$$, or $$${y}=\pm\sqrt{{{4}-{{x}}^{{2}}}}$$$. Because of $$$\pm$$$, we have two values of $$${y}$$$ for every value of $$${x}$$$. Also, as can be seen from the graph, the function fails the vertical line test.

Parametric representation of a function.

Parametric representation is a slightly changed form of representing a function by a formula.

When we define a function by a formula, we use the following record: $$${y}={f{{\left({x}\right)}}}$$$, where $$${f{}}$$$ is some formula.

But often, it is more convenient to represent a function using two functions and a parameter.

If we define a function as $$${x}={u}{\left({t}\right)},{y}={v}{\left({t}\right)}$$$, by changing the value of $$${t}$$$ (parameter), we can find the corresponding pairs of $$${x}$$$ and $$${y}$$$.

In general, we set some restrictions on $$${t}$$$. If we require $$${a}\le{t}\le{b}$$$, the point $$${\left({u}{\left({a}\right)},{v}{\left({a}\right)}\right)}$$$ is called initial, and the point $$${\left({u}{\left({b}\right)},{v}{\left({b}\right)}\right)}$$$ is called terminal.

In some cases (but not always), it is possible to eliminate the paremeter $$${t}$$$ and obtain the standard representation of $$${y}$$$ as a function of $$${x}$$$ (or $$${x}$$$ as a function of $$${y}$$$).

Also, note that parametric equations can generate curves that are not functions.

And one more final example.