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.