Properties of functions

A function is a rule that assigns to every number x in the domain M exactly one y in the range N. We usually write a function in the form y = f(x), or we can express it explicitly as f:y = x, where the variable x is the argument of the function.

Domain and range

For every function we also have to determine its domain, which is the set of all permitted values of the argument x, that is, all the values the variable x can take. We call this the domain of f.

A simple example: for f:y = x the domain is the whole set of real numbers, ℝ. Another example: for $f:y=\frac{1}{x}$ the domain is the set of real numbers again, but this time without zero, because you cannot divide by zero and the expression would make no sense: the domain is ℝ ∖ {0}.

The range is then, in the same way, the set of all possible y, that is, the set of all the values that the function f can take. Again a simple example: take the function f:y = x. Here the range is the set ℝ, because y can take any value in this set. Let’s take another function, though: f:y = |x| (modulus, or absolute value). Here the range is the interval [0, +∞), because y can never equal, say, minus five, since that is impossible by the definition of the modulus.

If we think of a mapping from a set A to a set B, then the set A is the domain and the set B is the set the function maps into (the codomain); the range is a subset of it.

Monotonicity of a function

Monotonicity is a property that tells us whether a function is increasing, decreasing, non-increasing, non-decreasing, constant or none of these on a given interval.

This property is best read from the graph: if you feel that the graph is going down, the function is decreasing, and if it goes up, the function is increasing. As simple as that :-). The definitions look like this: take a function f and let x₁, x₂ be in the domain of f. Then we say that

• the function f is increasing if x₁ < x₂ implies f(x₁) < f(x₂).
• the function f is decreasing if x₁ < x₂ implies f(x₁) > f(x₂). (The definition looks the same, but in the first case there is < between the function values and in the second >.)

Look at the graph of the increasing function f(x) = ln(x):

We can see that when we choose x₁, x₂ so that x₁ < x₂, it is also true that f(x₁) < f(x₂). Now look at another graph, this time of the function $f(x)=\frac{1}{x}$, which is decreasing, for example, on the interval (0, ∞):

Here we see that when x₁ < x₂, the opposite holds for their function values: f(x₁) > f(x₂). Although x₁ is smaller than x₂, it “produces” a larger function value.

Here are the remaining definitions:

• A function f is non-decreasing if x₁ < x₂ implies f(x₁) ≤ f(x₂). (The definition is similar to the one for an increasing function; we only allow the function to take a constant value over some stretch.)
• A function f is non-increasing if x₁ < x₂ implies f(x₁) ≥ f(x₂).
• A function f is constant if its range is a set with a single element, in other words if f(x₁) = f(x₂) for any two x₁, x₂.

An example of a non-increasing function is f(x) = |x|−x:

Notice that on the interval (−∞, 0) the function decreases, but on the rest it has a constant value (zero). Over the whole domain it is therefore a non-increasing function.

Even and odd functions

Some functions can, under certain conditions, be even or odd. It is easiest if you know the graph of the function, as you can tell from it most quickly. An even function is symmetrical about the y-axis, whereas an odd function is symmetrical about the origin (0, 0). An example of an even function is y = x² and an example of an odd function is y = 2x. The definitions are as follows – a function f is

• even if f(x) = f(−x) for all x in the domain of f,
• odd if −f(x) = f(−x) for all x in the domain of f.

Even and odd functions are covered in a separate article, Even and odd functions.

One-to-one functions

We recognise a one-to-one function by the fact that all its function values are unique: no function value repeats. Definition:

A function f is one-to-one if, for all x₁, x₂ in the domain of f,

$$x_1\ne x_2\Rightarrow f(x_1)\ne f(x_2)$$

If we choose two different values of x from the domain of the function, their function values must always be different.

Graphically this is, as always, quite easy to tell. If you draw a line parallel to the x-axis across the graph, it must cross the graph at most once. An example of a one-to-one function is the linear function y = 3x. Whichever line you take, it crosses the graph exactly once. The opposite is, for example, the sine function, whose graph forms a sort of wave, so if you draw a line parallel to the x-axis through the point $\frac{1}{2}$ on the y-axis, you find infinitely many points of intersection with the graph, and so the function is not one-to-one.

Bounded functions

Functions can be bounded above, bounded below, or bounded both above and below – we simply call such a function a bounded function. A function f is bounded above if we can find a real number A that is greater than all the values of the function f. Similarly for a function bounded below. So the definitions are:

• A function f is bounded above if there is A ∈ ℝ such that A > f(x) for all x in the domain of f.
• A function f is bounded below if there is A ∈ ℝ such that A < f(x) for all x in the domain of f.

Graphically we can picture it as looking for a line parallel to the x-axis (that is, a horizontal line) that lies above or below the whole graph of the function.

The picture shows the graph of the function f(x) = x⁴, which is bounded below. Notice that we do not have to find the A that is “closest” to the graph; it is enough to place the line anywhere below the graph.

Minimum and maximum

A function f has a maximum at a point M of the domain if the value at this point M is the highest of all the values the function takes. Similarly for the minimum. Definitions:

• A function f has a global maximum at the point M in its domain if f(x) ≤ f(M) for all x in the domain of f – in short, the function value is highest at M. If there is only one such point, we call it a strict maximum.
• A function f has a global minimum at the point m in its domain if f(x) ≥ f(m) for all x in the domain of f – the function value is smallest at m.

On a graph you can probably find the maximum and minimum of a function quite easily: you simply look at where the function is lowest or highest. Of course, a function need not have either a minimum or a maximum, or it may have only a minimum or only a maximum, and a function may also have only local extrema but no extremum over the whole domain.

If you need to find minima and maxima, use derivatives; the exact method is described in the article on curve sketching.

Example

Work out the properties of the function from the following graph:

• The domain of this function is the interval [−4, 4].
• The range is the interval [−2, 4].
• The function looks as if it might be increasing, but on the stretch [0, 2] it has a constant value, so it cannot be increasing. However, the function does not decrease on any interval, so it is a non-decreasing function.
• The graph of the function is not symmetrical about the y-axis or about the origin, so the function is neither even nor odd.
• Is the function one-to-one? We have already said that on the stretch [0, 2] the function has a constant value, so it cannot be one-to-one. For different values of x it has the same y.
• The function is bounded, both above and below. We can find a horizontal line that lies above the whole graph and one that lies below the whole graph.
• The given function also has a minimum and a maximum. It has its minimum at the point m = −4 and its maximum at M = 4.