Relations and functions worksheet with answers and solved examples
Definition of Relation
A relation is established between two sets when an object from one set is related to one object from another set, resulting in the formation of ordered pairs. For instance, x is an object of set A and y is an object from set B. Then they share a relationship and can be written as an ordered pair (x, y).
Definition of Function
To define, a function is a binary process or relation that makes each element of one set somehow related to exactly one element from the second one.
Now, let’s begin with the relations functions worksheet.
Practise Worksheet Relations and Functions
1. P x P is a Cartesian product having nine elements. Already known sets of ordered pairs are (-2, 0) and (0, 2). Find set P and the remaining ordered pairs.
2. 120 students are studying in class 11 divided among four sections. Now consider set A to be the students and B to be the sections. Form a relation from A to B to show whether the students belong to any of the four sections. Also mention whether this relation is a function or not.
3. Expression to convert Fahrenheit to Celsius is y = (5x/9) – (160/9). Evaluate the inverse and also rule out whether the same is a function or not.
4. Find out whether the following statements are true or false.
(a) Relation {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} is a function also.
(b) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)} is not a function.
(c) All functions are relations, but all relations are not functions.
(d) P = (1, 2, 3} and Q = {e, f}. P x Q provides number of relations to be 64.
(e) {(x,y) | x=3 and y is a real number} is both a function and relation.
5. R is a relation {(6,4), (8,-1), (x,7), (-3,-6)}. Find the value of x that will make this relation a function.
6. Consider n (A) = m and n (B) = n. Find out the number of relations (non-empty) that can be predicted from A to B.
7. f (x) = x3 – 1 / x3
What will be f(x) + f(1/x)?
8. Find the range and domain of a real function ‘f’ represented by f(x) = root over (x -1).
9. What will be the domain and range of the following relation?
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10. The following table contains data of a woman’s forehand with her respective height. With this data, a student maps a relationship between forehand and hand in the form y = ax + b, provided both are constants.
Determine whether this relation is a function or not, and find the values of both a and b.
11. If f (x) = 2x – 3, then find [f (0) + (f)] / 2 and value of x when (a) f(x) = 0 (b) f(x)= x and f(x) = f(1-x)
12. Consider an aeroplane moving at a velocity of 500 km per hour. Illustrate the travelled distance in the form of function t in hours.
13. Map the following relation and also mention the domain and range.
{(-2, 1), (0, 3), (5, 4), (-2, 3)}
Relations and Functions Worksheet Answer Key
1. Set P = {-2,0,2} and the remaining elements are {(-2,-2), (-2,2), (0,-2), (0,0), (2,-2), (2,0), (2,2)}
2. As all the 120 students belong to class 11, every pupil must be in any of the sections. So, this relation is a function as well.
3. f-1 (x) = (9x/5) + 32 and the inverse is a function also.
4. (a) True (b) False (c) True (d) True (e) False
5. 1
6. 2mn – 1
7. 0
8. Domain = (1, ∞) and Range = (0, ∞)
9. Range = {4, 16, 25, 36} and Domain = {-2, 2, 4, 5, 6}
10. As every value of x has a different value of y; it is a function. The value of a is 0.90 and b is 24.5
11. Value of [f(0) + f(1)] / 2 is -2, (a) x = 3/2 (b) x = 3 (c) x = ½
12. Required distance is 500t
13. Domain = {-2, 0, 5}, Range = {-1, 3, 4}
Mapping Figure:
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This relations functions worksheet may act as a last-minute tool to revise the function and relation problems before your exams. You must try to solve the questions on your own and later check it with the given practice worksheet relations and functions answer key.
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FAQs on Relations And Functions Worksheet For Practice And Mastery
1. What is a relation in mathematics?
A relation in mathematics is a set of ordered pairs that shows how elements of one set are connected to elements of another set. In a relations and functions worksheet, relations are usually written as:
- A set of ordered pairs, for example: {(1,2), (2,4), (3,6)}
- A table of values
- A mapping diagram
- A graph on the coordinate plane
The first value in each ordered pair is called the input (or domain element), and the second value is the output (or range element).
2. What is a function in mathematics?
A function is a special type of relation in which each input has exactly one output. This means:
- No input value is paired with more than one output.
- Each x-value corresponds to only one y-value.
For example, {(1,2), (2,4), (3,6)} is a function, but {(1,2), (1,3)} is not a function because the input 1 has two different outputs.
3. What is the difference between a relation and a function?
The main difference is that a function has exactly one output for each input, while a relation may have more than one output for the same input.
- All functions are relations.
- Not all relations are functions.
- If any x-value repeats with different y-values, it is not a function.
This distinction is commonly tested in relations and functions worksheets using tables, graphs, or ordered pairs.
4. How do you know if a relation is a function?
A relation is a function if each input value is paired with only one output value. To check:
- From ordered pairs: Make sure no x-value is repeated with different y-values.
- From a table: Ensure each input appears only once.
- From a graph: Use the vertical line test; if a vertical line crosses the graph more than once, it is not a function.
If any input maps to two outputs, the relation fails the function rule.
5. What is the domain and range of a relation?
The domain is the set of all input values, and the range is the set of all output values in a relation or function. For example, in {(1,2), (2,4), (3,6)}:
- Domain = {1, 2, 3}
- Range = {2, 4, 6}
To find the domain, list all first coordinates; to find the range, list all second coordinates without repeating values.
6. What is the vertical line test for functions?
The vertical line test states that a graph represents a function if no vertical line intersects it more than once.
- If a vertical line crosses the graph at only one point everywhere, it is a function.
- If it crosses at two or more points, it is not a function.
This test helps determine whether a graphed relation satisfies the definition of a function.
7. How do you write a function in function notation?
A function is written in function notation as f(x), which means “the value of function f at x.” For example, if y = 2x + 3, we write:
- f(x) = 2x + 3
To evaluate, substitute the input into the expression. If x = 4:
- f(4) = 2(4) + 3 = 8 + 3 = 11
Function notation clearly shows the relationship between input and output.
8. How do you find the value of a function for a given input?
To find the value of a function, substitute the given input into the function rule and simplify. For example, if f(x) = x² − 5 and x = 3:
- Step 1: Substitute 3 → f(3) = 3² − 5
- Step 2: Square 3 → 9 − 5
- Step 3: Simplify → 4
So, f(3) = 4. This process is commonly practiced in relations and functions worksheets.
9. Can a function have the same output for different inputs?
Yes, a function can have the same output for different inputs as long as each input has only one output. For example, in the function f(x) = x²:
- f(2) = 4
- f(−2) = 4
Both inputs give the same output, but since each input has exactly one output, it is still a function.
10. What are common mistakes in relations and functions worksheets?
Common mistakes in a relations and functions worksheet usually involve misunderstanding the function rule or misidentifying domain and range. Frequent errors include:
- Thinking repeated y-values mean it is not a function (only repeated x-values matter).
- Forgetting to apply the vertical line test correctly.
- Listing duplicate numbers in the domain or range.
- Substituting incorrectly when evaluating function notation.
Avoiding these mistakes helps ensure accurate identification and evaluation of relations and functions.
