Math Lesson 16.1.6 - Function

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Welcome to our Math lesson on Function, this is the sixth lesson of our suite of math lessons covering the topic of Relation and Function, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Function

So far, we have dealt with functions in many tutorials. From those tutorials, you will already know that a function is a kind of relation between two sets, where the elements contained in them are known as variables. Thus, the first set in a function is the set of independent variables X while the second set is that of the dependent variables Y. Each of these sets is represented visually through an axis in the coordinate (Cartesian) plane. However, although we have given a lot of information about functions (including some types of functions and their graph), we still don't know the exact definition of a function and how this concept relates to the other concepts we have provided earlier in this tutorial.

By definition, a function is a special type of relation, where each element of the independent variables set X has in correspondence a single element of the dependent variables set Y.

In other words, in a function an x-value has in correspondence a single y-value. The reverse is not a must, i.e. an y-value may have in correspondence more than an x-value (for example in y = x^x, where x may be positive or negative but when raised in the second power it gives always a positive y-value).

All relations discussed so far except one (in Example 4) were also functions. Let's consider an example to clarify this point.

Example 6

From the definition of function, find whether the following relations are also functions or not.

1. y = x^x - 5
2. y^x = -x^x + 4
3. y^x = x

Solution 6

The simplest way to prove that a relation is not a function is to provide a counter-example, i.e. to find an x value that has in correspondence two y-values.

1. This relation is a function because it is impossible to find an x-value that has in correspondence two y-values. We cannot substitute all numbers in the relation to prove this, but it is sufficient to pick up a positive number, a negative one and zero and substitute them in the formula, as the logic is the same for the rest of the numbers. Thus, for x = -2 we have
   y(-2) = (-2)² - 5
   = 4 - 5
   = -1 (a single y-value)
   For x = 0 we have
   y(0) = 0² - 5
   = 0 - 5
   = -5 (a single y-value)
   For x = 3, we have
   y(3) = 3² - 5
   = 9 - 5
   = 4 (a single y-value)
   Therefore, this relation is a function because all x-values have in correspondence a single y-value.
2. If we pick a random x-value (for example x = 1), the relation becomes
   y² = -1² + 4
   y² = -1 + 4
   y² = 3
   We have two y-values that give 3 when raised in the second power. They are -√3 and √3. Therefore, this relation is not a function because we identified an x-value which has in correspondence two y-values.
3. Let's pick a value, for example, x = 9. Thus, we obtain
   y² = 9
   This equation is true for two x-values: -3 and 3, as both give 9 when raised to the second power. Therefore, this relation is not a function.

You have reached the end of Math lesson 16.1.6 Function. There are 9 lessons in this physics tutorial covering Relation and Function, you can access all the lessons from this tutorial below.

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2. Functions Math tutorial: Relation and Function. Read the Relation and Function math tutorial and build your math knowledge of Functions
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6. Continuing learning functions - read our next math tutorial: Injective, Surjective and Bijective Functions. Graphs of Functions

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