Properties of numerical functions

In many sciences, you come across various functions. In most cases, they are used to describe a dependency between two values you can measure, mostly numbers. For example, the dependency between the voltage and current through a conductor (Ohm's law), or the relationship between unemployment and losses in a country's production (Okun's law). In this topic you'll learn the properties of numerical functions – they are of great help if you need to analyze any process described by a function.

What is a numerical function?

A numerical function is a function that takes on values in some set of numbers – for example, in the set of integers $\mathbb{Z}$ or the set of real numbers $\mathbb{R}$. A numerical function does not obligatorily have a set of numbers as a domain (the set of its possible inputs), but its codomain (the set of possible outputs) is always a set of numbers.

For example, a relation that maps a real number with its tripled value is a numerical function: $f(x) = 3x$. In this example the role of both the domain and the codomain is played by the set of real numbers: $f \colon \mathbb{R} \to \mathbb{R}$. You may come across other numerical functions, e.g. $g \colon \mathbb{Z} \to \{0,1 , 2\}$ such that $g(x)$ is the remainder when $x$ is divided by 3. In this example, the domain is the set of integers, and the codomain is the set $\{0, 1, 2\}$. And here's an example of a numerical function whose domain is a set of letters. This function maps the set of the English alphabet letters with their ordinal numbers: $a \to 1 , b \to 2 , \dots, z \to 26$.

How to define a function

Defining a numerical function $y = f(x)$ means specifying a rule that allows one to calculate the corresponding value of $y$ from an arbitrarily chosen value of $x$. Most often, this rule is given by the formula. For example: $f(x) = 5x + 2 + x^2,$or$g(x) = \dfrac{x+1}{2x^2+5} \ \text{with domain } \ (-2,4].$The domain restriction means that we require $-2 < x \leqslant 4$ in order for $g(x)$ to be defined (the round bracket indicates that $-2$ is not included in the domain, and the square bracket indicates that $4$ is included ).

Sometimes functions are defined piecewise – using different output formulas for different pieces, or parts, of the domain. For example,

$h(x) = \begin{cases} -x \text{ for } x < 0 \\ x^{2} \text{ for } 0 \leq x \leq 3\\ 1 \text{ for } x > 3 \end{cases}$

If our function $f$ is defined on the set of real numbers or its subset, we can draw a graph of this function – that is, all points on a plane with coordinates $(x, f(x))$. Let's see, what the graph of the function $f(x) = 5x + 2 + x^2$ looks like:

Graphs of functions can be much more complicated. For example, the graph of our piecewise function $h(x)$ that we wrote above looks like this:

Boundedness

A function $f$ is

• bounded above if there is a number $M$ that is greater than or equal to every other number in the range of $f$;

• bounded below if there is a number $m$ that is less than or equal to every other number in the range of $f$;

• bounded if it is bounded above and below.

For example, the function $f_{1}: \mathbb{R} \rightarrow \mathbb{R}$ such that $f_{1}(x) = x^{2}$ is bounded below (because every value of $f_{1}$ is greater than or equal zero), but it's not bounded above. Note that the function $f_{2}(x) = -x^{2}$ is bounded above but not below.

The function $f_{3}(x) = \sin(\pi x / 2)$ is bounded both above and below:

On the set of real numbers, we can define the function$sgn(x) = \begin{cases} -1 \text{ if } x < 0\\ 0 \text{ if } x = 0\\ 1 \text{ if } x > 0\\ \end{cases}$

known as the sign function. It is bounded as it can't take values greater than 1 or less than -1.

Increase and decrease

To define increasing and decreasing functions, we must have a numerical domain, for example, the set $\mathbb{R}$ of real numbers.

A function $f \colon X \to Y$ is called increasing if$f(x_2) \geqslant f(x_1) \ \ \text{for all} \ \ x_2 > x_1$. If $f(x_2) > f(x_1)$ for all $x_2 > x_1$ the function is called strictly increasing.

In other words, the value of an increasing function increases as its argument increases.

A function $f \colon X \to Y$ is called decreasing, if$f(x_2) \leqslant f(x_1) \ \ \text{for all} \ \ x_2 > x_1$. If $f(x_2) <f(x_1)$ for all $x_2 > x_1$ the function is called strictly decreasing.

In other words, the value of a decreasing function decreases as its argument increases.

We can notice that the function $sgn(x)$, introduced in the previous section, is increasing but not strictly increasing. On the other hand, the function $f_{4}(x) = -\sqrt{x}$, defined on non-negative real numbers, is strictly decreasing.

Sometimes a function can be increasing or decreasing on a piece of the domain. A set of real numbers lying between two fixed numbers $a$ and $b$ is known as an interval. An open interval, denoted by $(a, b)$, does not include its endpoints:

$(a, b) = \{x \mid a < x < b\}$ - A closed interval, on the other hand, includes $a$ and $b$:

$[a, b] = \{x \mid a \leq x \leq b\}$ - We can look again at the graph of a function $f_{3}(x) = \sin(\pi x / 2)$ and notice that it is increasing on an interval $[-1, 1]$ and decreasing on an interval $[1, 3]$.

Symmetry

A function $f \colon X \to Y$ is called an even function, if $f$ satisfies the following condition:$f(x) = f(-x) \ \ \text{for all } \ x \in X.$This means that $f$ is symmetric about the y-axis.

For example, $f_{5}(x) = x^6 -10$ is even:

A function $f \colon X \to Y$ is called an odd function, if $f$ satisfies the following condition:$f(-x) = -f(x) \ \ \text{for all } \ x \in X.$This means that $f$ is symmetric if we rotate it by $180^\circ$ about the origin. For example, $f_{6}(x) = x^5$ is odd:

A function can be either even, odd, or neither, with the exception of $f(x)=0$, which is both even and odd. The function $f_{7}(x) = 0.5x + \cos(x)$ is neither even nor odd:

Conclusion

Numerical functions are very often used in practice to describe different dependencies. You can understand the nature of these dependencies in terms of the functions' properties. Now you can define such classes of functions as:

• bounded and unbounded

• increasing and decreasing

• even and odd

For example, the function describing the number of solved problems from the time spent on the Hyperskill platform will be bounded (you can't solve less than 0 problems and more than all problems) and increasing (once you solve a problem, you can't unsolve it).