Range of Values: Determine, Definition & Functions

In this article we want to explain everything about the value range and answer all your questions about it. The range of values ​​is a topic of the curve discussion and is taught in mathematics.

range of values definition

The range of values ​​can also be called a set of values. You can use the range of values ​​to determine which y-values ​​a function takes on. The value range of a function f(x) is also marked with .

! The range of values ​​answers the question: “What y-values ​​does the function f take?” !

General example of the range of values

As an example we examine the function f(x) = x². The domain of definition is given and contains = {1,2,3,4,5}. This means that the domain of definition tells you that you can only use the values ​​1, 2, 3, 4 and 5 in the function f(x) = x². The domain is thus the set of y-values ​​you get after substituting each x from the domain into the function.

Let’s insert the values ​​from the domain:

f(1) = 1² = 1

f(2) = 2² = 4

f(3) = 3² = 9

f(4) = 4² = 16

f(5) = 5² = 25

The numbers in bold are the values ​​for the value range. Accordingly, the value range is: ={1,4,9,16,25}.

Value range linear function Determine and specify

As you should already know, linear functions are defined throughout R. That means you can substitute any real number for each x of a linear function. As a result, linear functions assume every y-value. Thus, for the range of values: = R.

To understand it better, we have prepared an example for you.

Example 1: Value range of a linear function

The graph of the function f(x)= x+2 is given.

The domain of the function is as follows: = R

The range of the function is: = R

Source: mathebibel.de

In the task, the definition range of a function can be restricted as desired. Now, if we look at the example above: f(x) = x+2, let’s assume that the domain is limited to = {0;2}.

How do you calculate the range of values ​​now?

Very simple: First you put the lower limit of the interval (0) into the function to find the smallest y-value. Then you plug the upper bound of the interval (2) into the function to get the largest y value:

f(0) = 0+2 = 2

f(2) = 2+2 = 4

The smallest y-value (2) and the largest y-value (4) are the limits of the searched value range. Thus: = {2,4}

Viewed graphically, the domain of definition (all permitted x-values) corresponds to the x-axis and the range of values ​​(all possible y-values) can be read off from the y-axis.

Range of quadratic functions

As you should already know, quadratic functions are defined throughout R. That means you can substitute any real number for each x of a linear function. But unlike linear functions, quadratic functions don’t always take on every y-value.

The following therefore applies to the value range of a quadratic function:

  • if the sign of x² positive is
  • if the sign of x² negative is

where is the coordinate of the vertex.

In the next example, you should already know how to compute vertex

We determine the set of values ​​with the following calculation steps:

  1. Read the sign of x²
  2. Calculate vertex
  3. Determine range of values

Example 1: Value range of quadratic functions

Let the graph of the function f(x) = x²-6x+10 be given.

The domain of the function is = R.

The apex of the parabola is at S (3 |1).

For the value range = .

Source: mathebibel.de

Example 2: Range of quadratic functions

Given the graph of the function f(x) = -x² +8x -14.

The domain of the function is .

The apex of the parabola is at S (4 |2).

For the value range = .

Source: mathebibel.de

The limits for the range of values ​​of quadratic functions depend on two factors:

  • y – coordinate of the vertex
  • sign of x²

Why?

The graph of a quadratic function is a parabola. And the apex the parabola is the point where the graph of the function den highest y-value (= high point HP) or the lowest y-value (= low point TP) assumes To find out if it’s an HP or a TP, you just have to look at the sign of x² of the function. That’s how you’ll know.

Range of values ​​of special functions

In order to be able to determine the value range of a function, in most cases you also have to calculate the extreme points, i.e. high points and low points, and carry out a limit value analysis. Therefore, the determination of the value range is often part of the curve discussion.

You can find out more about the curve discussion of special functions in our articles on the topic of curve discussion. Enjoy reading!

range of values All important at a glance

In summary one can say:

  • The range of values ​​shows you what possible y-values ​​there are for a function.
  • In the case of linear functions, all real numbers can be used as a range of values.
  • The domain limits the x-values ​​that can be used.
  • For quadratic functions, you can tell what the range of values ​​looks like by looking at the sign of x² and the y-coordinate of the vertex.

Well done! After diligently reading through everything, you should now know everything about the value range. 🙂 Keep it up!