Function and its property for KS4

If two quantities are interdependent, we can write an expression for them, which we mathematically call a function. The function can be presented in different ways.

The two quantities may be interdependent; one is called a dependent variable and the other an independent variable:

We also say that the dependent variable y is a function of the independent variable x, and we write this with

where f is a rule and f(x) is the value of this function given x.

Function or mapping

A function is a rule that assigns exactly one element from the set B to each element x from the set A. Graphically, this can be illustrated with an arrow diagram.

We cannot talk about a function when:

We will agree on the following labeling:

The Domain is the set on which the function is defined. The elements are also called the original.

The range is the set of all function values. The set can be the same as the set in which the rule is or just a subset of it.

The function can be presented in different ways. Most often we present it with a rule. In case the domain and the range are finite sets, the function can be represented by a table or by an arrow diagram. Real functions can also be represented by a graph.

The function can be given by telling how the function maps images. This is stated in the rule:

The function can be given with a table in the case where the domain and the range are finite sets. In the first line, enter the values of the independent variable, and in the second, the values of the dependent variable.

However, if we are talking about a real function of a real variable, the mapping cannot be given entirely with a table. The table can only illustrate some of the values of the function rule.

First, we draw two diagrams to illustrate the set A and the set B. We use the arrows to illustrate which elements of the set A are mapped to the elements of the set B.

A graph of a function is the basic way to represent a function in mathematics. The graph of the function f is defined as a set of ordered pairs (x, y), where the first element x passes the entire domain of the function, and the second element y is the image of the corresponding x: so the relationship applies between the components. Mathematically, we write this as:

what we read: gamma is the set of all pairs (x, y), where x is the element of the set of the domain , and y is the mapping of the element x, according to the rule f.

The ordered pairs (x, y) can be drawn in the coordinate system. When the set A is finite (has a finite number of elements), then to each element (x, f (x)) belongs in the coordinate system a point whose abscissa is x, ordinate y so its image is f (x).

In the following chapters, we will learn to find the the intercepts on the x and y axes.

The x-intercept of a function is the number x for which the value of the function f is equal to 0. The graph of the function intersects the abscissa axis at the value of the variable x for which the function has a value of zero. If we are looking for a computational zero, we find the value xwith the rule:

In the linear equation occurs x to the first power. For this reason, the line is also said to be a first-degree equation; its zero is also called simple zero. The graph of a linear function intersects the abscissa axis at a certain nonzero angle.

In the case of equations of higher powers - these are the equations where the variable x occurs at higher powers (see the chapter polynomial) - we can also talk about the degree of zero. We say that is a zero of degree if we can write the equation in the form:

where .

If n is equal to 2, 4, 6, ... we say that the zero is an even degree. If n is equal to 1, 3, 5, 7 ... we say that the zero is an odd degree.

In the following we will look at the behavior of the function in odd levels of function.

In even zeros, the graph of the polynomial of the abscissa axis only touches and does not cross it.

At zeros of odd degree, the graph of the polynomial fits very well to the abscissa axis and also crosses it.

y-intercepts is the point where the graph intersects the ordinate (y) axis. Since the point lies on the ordinate axis, its abscissa is equal to 0, and the ordinate is calculated by inserting 0 for the value x:

The point is therefore of the form: .

Injectivity, surjectivity, and bijectivity are properties of a function. Let's take a look at them.

The function is injective when it maps different values into different images.

We check the injectivity in the graph by drawing parallels to the abscissa axis. In the case where the function is injective, each such parallel intersects the graph at only one point. If the line graph intersects at two (or more) points, then the function is not injective.

A function is surjective if its range is equal to the set B. In other words, if each element from the set B is an image to at least one element from the set A.

If a function is injective and surjective at the same time, we say that the function is bijective. In other words, each element from the set B is a picture of exactly one element from the set A.