Math Lesson 16.1.9 - Domain and Range of a Function

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

Welcome to our Math lesson on Domain and Range of a Function, this is the ninth 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.

Domain and Range of a Function

Another restriction in the number of the ordered pairs that may be solutions for a given function may be the limitation in the choice of the x-values. For example, the exercise may require finding the images of f(x) = 3x - 1 from x = 1 to x = 4. In this way, for x = 1 we find f(1) = 3 · 1 - 1 = 3 - 1 = 2 and for x = 4 we find f(4) = 3 · 4 - 1 = 12 - 1 = 11. Therefore, since we have a limitation in the input, i.e. in the possible choices of the x-values, there is a limitation in the output values (y-values) as well. Hence, the ordered pair (5, 14) does not belong to the given function, despite it gives a true result when substituted in the function's formula (3 · 5 - 1 = 14). This is because x = 5 is outside the set of the values determined by the initial conditions (x must be between 2 and 4). As a result, the corresponding y-value will be outside the range of the images as well. Therefore, an ordered pair not only must give a true result when substituted in the function's formula but also it must be within the allowed values determined by the initial conditions of the function.

By definition, the set of the allowed values for the x-variable in a function is known as the domain D. In our example, we have D = [2, 4]. (Recall the symbols of segment and interval explained in chapter 10.)

On the other hand, the set of images resulting from the substitution of the input values from the domain D in the formula of a function is called the range R. In our example, we have R = [2, 14] as they are the minimum and maximum values of the function for the given domain.

Example 8

Find the range R of the following functions if the domain for all of them is D = [-1, 4].

1. f(x) = 3 - 2x
2. f(x) = 1/(x + 2)
3. f(x) = 4 - x^x

Solution 8

We substitute the biggest and the smallest number from the domain to obtain the minimum and maximum values of the range (it does not always work this way though!). Hence, we have

1. For x = -1:
   f(-1) = 3 - 2 ∙ (-1)
   = 3 + 2
   = 5
   and for x = 4:
   f(4) = 3 - 2 ∙ 4
   = 3 - 8
   = -5
   Therefore, the range of this function is R = [-5, 5].
2. For x = -1:
   f(-1) = 1/(-1) + 2
   = 1/1
   = 1
   and for x = 4:
   f(4) = 1/4 + 2
   = 1/6
   Therefore, the range of this function is R = [1/6, 1].
3. For x = -1:
   f(-1) = 4 - (-1)²
   = 4 - 1
   = 3
   and for x = 4 :
   f(4) = 4 - (4)²
   = 4 - 16
   = -12
   In this case, we must also consider the vertex of the parabola, as it may be inside the domain. Since this is a quadratic function, we check for the roots by substituting f(x) = 0.
   f(x) = 4 - x²
   0 = 4 - x²
   x² = 4
   Thus, the roots are x¹ = -2 and x^x = 2. The x-coordinate of the vertex is in the midpoint of these two roots, i.e.
   x_V = x₁ + x₂/2
   = -2 + 2/2
   = 0/2
   = 0
   Hence, the y-value of the vertex is
   f(0) = 4 - 0²
   = 4
   Therefore, the range of this function in the given domain is not [-12, 3] as it would result if considering only the endpoints of the domain but we have R = [-12, 4] instead.

If the domain is not explicitly given in the clues, we take it as R (the set of real numbers; do not confuse it with the R of range).

You have reached the end of Math lesson 16.1.9 Domain and Range of a Function. There are 9 lessons in this physics tutorial covering Relation and Function, you can access all the lessons from this tutorial below.

More Relation and Function Lessons and Learning Resources

Whats next?

Enjoy the "Domain and Range of a Function" math lesson? People who liked the "Relation and Function lesson found the following resources useful:

1. Domain Range Feedback. Helps other - Leave a rating for this domain range (see below)
2. Functions Math tutorial: Relation and Function. Read the Relation and Function math tutorial and build your math knowledge of Functions
3. Functions Revision Notes: Relation and Function. Print the notes so you can revise the key points covered in the math tutorial for Relation and Function
4. Functions Practice Questions: Relation and Function. Test and improve your knowledge of Relation and Function with example questins and answers
5. Check your calculations for Functions questions with our excellent Functions calculators which contain full equations and calculations clearly displayed line by line. See the Functions Calculators by iCalculator™ below.
6. Continuing learning functions - read our next math tutorial: Injective, Surjective and Bijective Functions. Graphs of Functions

Help others Learning Math just like you

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

We hope you found this Math tutorial "Relation and Function" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.

Functions Calculators by iCalculator™