What is a math function

Functions in Mathematics

In the last article Relations and Functions, we used an example to see that a relation is a pair in which the elements are formed based on an assignment rule. In addition, in mathematics we mean one function an at least unambiguous relation. In this post, I'll first introduce a few examples and exercises for functions. After that, I'll show you how to make one Table of values created for a function. For this I am hiring interactive JavaScript available for this. Then I explain what types of functions there are and which mathematic spelling to. Then I ask Examples of mathematical functions and functional equations based on their graphs in front. Finally I show them definition the Definition set D and the set of values ​​W. Last I give Tips on how to perform tasks related to functions.

When considering natural, technical or everyday processes mathematically, the value of one variable often depends on the value of another variable. For example, a car's gasoline consumption depends on its speed.

Examples of functions:

The price a commodity depends on the sold quantity from.
The grade a math thesis depends on the achieved score from.

Exercise 1 for functions:

Find more examples of such dependencies!
Possible solutions to this can be found below.

In other words: two sizes are assigned to each other.

quantity Price The quantity sold affects the price.
score Grade The number of points achieved in an exam influences the grade.

exercise 2 for functions:

Formulate the connections for your examples!

Definition of dependent and independent variables:

The sizes assigned to one another are called variables.
It is important to consider what dependency there is between the variables.

From the teacher's point of view, the grade for a math paper depends on the number of points achieved. However, the number of points does not depend on the grade.
The note is therefore also referred to as dependent variable and the score as independent variable.
In other words, there is a more functional Relationship between the number of points and the grade.

There are several ways of describing or displaying the functional relationship between two variables.
Just the example of the braking distance of a car.

How to create a table of values:

The braking distance is measured for different speeds and converted into a Table of values registered.

Create a table of values ​​for a function
Interactive: With this JavaScript you can create a value table for completely rational functions up to the 4th degree.

If you enter the table values ​​in a coordinate system and connect the points with each other, you get the Graph the function. This describes the relationship between speed and braking distance. Then you can find the values ​​of the independent variable x on the horizontal axis. The values ​​of the dependent variable y = f (x) can be found on the vertical axis. The horizontal axis will too abscissa called the vertical Ordinate.

Exercise 3:

Enter the points in a suitable coordinate system according to the table of values ​​and connect them to a graph.
Determine by reading the braking distance for speeds 30; 50; 70; 90 and 110 km / h.
If the police measured a braking distance of 90 m, how fast was the vehicle?

It is often also possible to represent the functional relationship using a function equation.
The following applies to the braking distance: y = f (x) = 0.01 x2.
Here y = f (x) stands for the braking distance and means that the y coordinate in the coordinate system depends on the variable x. In other words, the y-coordinate is a function of the independent variable x.
f (x) = 0.01 x2 is the function equation which specifies the rule how the values ​​for the dependent variable f (x) are to be formed.
In other words, f (x) = 0.01 x2 is the Functional rule.

Exercise 4:

Calculate the readings if the functional equation f (x) = 0.01 x2 reads.

After what has been worked out so far it can be said:

Definition of function in mathematics:

For a function, any value of the independent variable x exactly one Function value assigned to f (x). In other words, a function is a clear assignment, in which exactly one function value f (x) is assigned to an independent variable x from the definition set D. The functional relationship is described by a function equation. By inserting x values ​​into the function equation, function values ​​are obtained which can be displayed together with the x values ​​in a table of values. Each pair of values ​​in the table corresponds to exactly one point in the Cartesian coordinate system. In many cases, the points created in this way can be connected to form a graph. The set of all x values ​​that can be used in the function equation is called the definition set. The set of all function values ​​that arise thereby belong to the set of values ​​W of the function.

If I speak of a function, then I only give the function equation to describe it, so only f (x) = ... ..

The name of the function is f.
The notation x → f (x) makes it clear that a certain function value belongs to every x value.
The function equation f (x) = x - 1 specifies the calculation rule for how the function values ​​are formed.

To show the uniqueness of the assignment again, the following is an example:
Every student has a certain shoe size. It does not matter that several students have the same shoe size.
The uniqueness relates to pupil → shoe size.
Expressed in terms of variables, this means that the student's name forms the independent variable and the shoe size assigned to him forms the dependent variable.
Conversely, the assignment of the shoe size to students is not clear, because several students can z. B. have shoe size 42.
A function of this kind has no function equation, but it can be represented as a set image.


Examples of mathematical functions and functional equations


Linear function (straight line)


Quadratic function (parabola)

Rational 3rd degree function

Totally rational 4th degree function

Exponential functions

Root function


Logarithmic functions


Trigonometric functions


exponential decay


exponential saturation curve


Hyperbola point symmetrical


Axisymmetric hyperbola


Definition of Definition set D

The Definition set of a function is the set of all independent variables for which the function is defined.

Definition ofSet of values ​​W

The set of values ​​of a function is the set of all function values ​​that arise from the elements of D.

Example 1:

Example 2:
It is not allowed to divide by zero.

Example 3:
The function can have any value.

Example 4:
Only positive values ​​appear as function values.


Mathematical notation for function equations and their meaning

For the value of the independent variable x = 4, the function value is 7.
The value pair (4 | 7) forms the coordinates of the point P (4 | 7) in the coordinate system.

The x-value to which the function value 9 belongs is searched for.

y is the function value for x = 5.


Summary

An unambiguous assignment in which exactly one function value f (x) is assigned to an independent variable x from the definition set D is called a function.

The functional relationship is described by a function equation (e.g. f (x) = 2x + 1).
By inserting x values ​​into the function equation, function values ​​are obtained which can then be displayed together with the x values ​​in a table of values.

Each pair of values ​​in the table corresponds to exactly one point in the Cartesian coordinate system.

In many cases, the points created in this way can be connected to form a graph.
The set of all x values ​​that can be used in the function equation is called the definition set.
The set of all function values ​​that result from this belong to the set of values ​​W of the function.


Solutions to the exercises

Exercise 1:

Find more examples of such dependencies.
Solution: For example, you might find these:
The power an internal combustion engine depends on the rotation speed from.
The surface a circle depends on his radius from.
The Electricity bill depends on the constant kWh price Amount of energy from.

Exercise 2:

Formulate the connections for your examples!
Solution:

rotation speed Power The speed of an internal combustion engine influences the engine power.
radius Area The radius of a circle affects its area.
Amount of energy Electricity bill The amount of energy used affects the electricity bill.

Exercise 3:

Enter the points in a suitable coordinate system according to the table of values ​​and connect them to a graph.
Determine by reading the braking distance for speeds 30; 50; 70; 90 and 110 km / h.
If the police measured a braking distance of 90 m, how fast was the vehicle?
Solution: For the speed of
At 30 km / h the braking distance is about 9 m.
At 50 km / h, the braking distance is around 25 m.
At 70 km / h, the braking distance is around 48 m.
90 km / h the braking distance is about 80 m.

For a speed of 110 km / h, the braking distance is around 120 m.
With a measured braking distance of 90 m, the vehicle drove at a speed of around 95 km / h.

Tips on how to do tasks on roles:

  • The x-axis (abscissa) of the coordinate system represents the set of independent variables, the y-axis (ordinate) that of the dependent variables.
  • Before value pairs are entered as points in a coordinate system, one must consider which is the independent variable and which is the dependent variable.
  • It is also important to find a suitable scale for scaling the axes.
  • To do this, look at the largest and smallest values ​​in the table and decide how many parts the two axes are to be divided into.
  • If you have then entered all points in the coordinate system according to the table of values, you connect these points to a graph.
  • You can then read off the corresponding function values ​​more or less precisely from the graph, even for those x values ​​that do not appear in the table of values.
  • You can also read off the associated x-value for a specific function value. The accuracy of the results depends on how accurately the graph was drawn.

Exercise 4:

Calculate the readings if the functional equation f (x) = 0.01 x2 reads.
Solution:
The function values ​​are calculated by plugging the corresponding x values ​​into the function equation.

At a speed of
At 30 km / h the braking distance is 9 m.
At 50 km / h the braking distance is 25 m.
At 70 km / h the braking distance is 49 m.
90 km / h the braking distance is 81 m.
110 km / h the braking distance is 121 m.

A braking distance of 90 m was measured, the speed is sought.

Only the positive value makes sense in relation to the task, since we only consider positive speeds. With a measured braking distance of 90 m, the speed was about 94.9 km / h.

Comment:
Only the calculation provides exact values, but it makes sense to check the calculated values ​​by reading them on the graph.

Here you will find tasks for functions II

and here tasks for functions VII

Here you will find an overview of other contributions to linear functions, it also contains links to other tasks.

You can buy these and other teaching materials in our shop. There teachers will find WORD files that they can change as required. You can also download all materials as PFD files for free.

In the next article Function and Inverse Function, the topic is deepened.