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Key Highlights
- A function indicates the relationship between a set of inputs having one output each.
- It is a special form of relation where the x-value should be associated with only one y-value.
- Domain, codomain, and range are part of sets that consist of all the elements of the ordered pairs of functions and relations.
- It can be illustrated in various forms, such as algebraic, set-builder, roster, and tabular.
- The domain is the set of values that we take as input, and the range is the set of values that we obtain as output.
- The range of a function is the real output of that function.
Domain, codomain, and range functions are sets of all possible values commonly used in functions and relations. In mathematics, a function within a specified range is referred to as a domain.
- In simpler terms, the domain can be defined as the set of values that are the input of the function.
- The range of a function refers to the set of all its outputs.
- Codomains are a set in which all the output of the function is constrained to fall.
- Here, we will discuss the domain, codomain, and range of a function, along with some important questions.
| Table of Content |
Key Terms: Domain, Codomain and Range, Functions, Domain, Range, Codomain, Relation, Cartesian Product
Functions
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Functions are among the most important tools in the construction of mathematical models. They specify the relationship between an element of one non-empty set and one element of another non-empty set.
- It is a relation that relates an input to an output.
- The concept is used to formulate, interpret, and solve major and minor real-life problems.
- A function is a relation where every element in the domain is related to another element in the codomain.
- It is often expressed as y = f(x), as a result of which f: A → B is a function such that (a, b)∈f for every a∈A and b∈B.
- One to One, Many to One, Onto, Into and Constant are different types of functions.
- Relations and functions can be represented in tabular form, arrow representation, algebraic form, set-builder form and roster form.
Real Life Application of FunctionsExample: From super-fast cars to mega skyscrapers are different applications of mathematical functions. |
Relations
According to the Cartesian product, two sets, A and B, should be the collection of all the possible pairs of ordered sets. In this way, a relation relates an element of one set to another element of some other set.
- Thus, the function is also a relation which is relating input to output.
- Mathematically, it can be represented as:
R = {(x,y): x ∈ A and y ∈ B}
- Where A: Domain of a function
- B: Range of a function
Some conditions that distinguish a function from any other relation, which further helps to define a function, are as follows:
- A function may not satisfy all mathematical values.
- It can be defined with the help of sets.
- A function relates an element of one set from exactly one element of another set and not more than this.
- On the other hand, a relation can relate to more than one element.
- Thus, a function is a unique kind of relation.
Example of RelationsExample: The relation R = {(9,3), (9,-3), (16, 4), (16, -4), (25, 5), (25, -5)} and it is represented using {}.  Relations |
Domain and Range
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The domain and range of a function can be explained with the help of functions in the real set.
Domain
The domain of a function is the sum of all the inputs or possible inputs to that function. It is usually detected when the denominator of the fraction is not equal to zero and the digit under the square root bracket is positive.
- It indicates all the values which go into a function.
- The domain of an exponential and logarithmic function is R and x>0, respectively.
Domain
How to find the Domain of a function
To find the domain, one needs to look at the values of the independent variables which are allowed to use independently. In other words, there should not be any zero at the bottom of the fraction and the square root should be free of any negative sign. In general, all real numbers in any set are regarded as the domain of a function. Though, some exceptions are always there.
Example of How to Find the Domain of a functionExample 1: When the given function is of the form f(x) = 3x + 9 or f(x) = 4x2 – 4, the domain is the set of all real numbers. Example 2: When the given function is of the form f(x) = 1/(x – 9), the domain is the set of all real numbers except 9. |
Range
All the real outputs of a function are called the range of the function. It refers to two closely related concepts: the codomain of the function or the image of the function.
- First, solve for x and consider the denominator ≠ 0 to determine the range of a function.
- For a surjective or onto function, the value of the codomain and the image of a function are the same.
- The range of a square root and an exponential function is y ≥ 0 and y>0, respectively.
Range
How to find the Range of a Function
Let's take an example of function X = f(y).
- The set of all the values of y from minimum to maximum indicates the range of the function.
- In the given expression, substitute different values of x to determine whether it is positive, negative or equal to other values.
- Determine the minimum and maximum values of y.
Codomain
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Codomain is part of a function representing a set of values, including range and set of additional values. It is sometimes equal to the range of the function. However, in certain cases, the range is the subset of the codomain.
- To put it in other words, all the values that have the possibility of becoming an output are known as codomains of a function.
- The range can be redefined by limiting the value of these functions.
| Example: Let's look at an image to better understand the domain, codomain, and range of a function when input and output values are provided.  Domain, codomain and range functions As apparent in the image, the function relates one element of a set (domain) to exactly one element of another set (range and codomain). Therefore, the domain is what can go into a function, i.e., the possible inputs.
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Sample Questions
Ques: Find the domain and range of a function f(x) = (3x)2 – 5? (3 marks)
Ans: Function is already given there. Let us find out range and domain of the function one by one.
function f(x) = (3x)2 – 5
Known: the domain of a function is the input values for f. On the other hand, the real outputs are the range functions.
Let y= (3x)2 – 5
(3x)2= y+5
(x)2= (y+5)/3
x= sqrt(y+5)/3
So, sqrt[(y + 5)/3] is bigger than 0.
This is possible only when y is bigger than y ≥ -5.
Hence, [-5, ∞) are the range of function f(x).
Ques: Find the domain and range of a function f(x) = (2x – 1)/(x + 4)? (4 marks)
Ans: The function is already provided.
function f(x)= (2x-1)/(x+4)
Known: the domain of a function is the input values for f. Moreover, the real outputs are the range functions.
In that case, we cannot define the function when x + 4 = 0, i.e. x = -4
Therefore, the domain of the provided function is the set of all real numbers except -4. Now, we will get at the range function.
Let y = (2x – 1)/(x + 4)
xy + 4y = 2x – 1
2x – xy = 4y + 1
x(2 – y) = 4y + 1
x = (4y + 1)/(2 – y)
This is interpreted only when y is not equal to 2.
Therefore, (-∞, 2) U (2, ∞) are the range of the given function.
Ques: If f(x) = (a – x)1/n, a > 0 and n ∈ N, then, what is the value of f(f(x))? (3 marks)
Ans: Function is already provided, f(x) = (a – x)1/n
Now, f(f(x)) = [(a – f(x))n]1/n
⇒ f(f(x)) = [(a – {(a – xn)1/n }n ]1/n
⇒ f(f(x)) = [a – (a – xn)]1/n
⇒ f(f(x)) = [a – a + xn)]1/n
⇒ f(f(x)) = (xn)1/n
⇒Therefore, f(f(x)) = x
Ques: A function f(x) is said to be an odd function if: (1 mark)
(a) f(-x) = f(x)
(b) f(-x) = -f(x)
(c) f(-x) = k * f(x) where k is a constant
(d) None of these
Ans: (b) f(-x) = -f(x)
Explantation: A function f(x) is said to be an odd function if
f(-x) = -f(x) for all x. Because real numbers are not of negative value. Furthermore, domains are always in positive values. Therefore, f(-x) = -f(x) is the answer.
Ques: The domain of the function f = {(1,3), (3,5), (2,6)} is… (1 mark)
(a) 3 and 2
(b) {1, 3, 2}
(c) {3, 5, 6}
(d) 3, 5 and 6
Ans: (b) {1,3,2}
Explantation: As discussed above, the domain is always a real number. According to the Cartesian Product, there is a set of (a,b). In that case, {1,3,2} are the set of a. Thus, we can identify them as domains.
Ques: The point on the curve y = x2 which is nearest to (3, 0) is…? (1 mark)
(a) (1, -1)
(b) (-1,1)
(c) (-1,-1)
(d) (1,1)
Ans: (d) (1,1)
Explantation: 1 as a real number in positive value is the nearest to (3,0) on the curve y=x2. It will be the nearest to both 3 as well as 0 when compared to other options and negative numbers.
Ques: Find the domain and range of the function f(x)= 1/ x−3? (3 marks)
Ans: function is already provided; f(x)= 1/ x−3
x-3 can not be equal to 0. Because it is not accepted as a domain.
Hence, the Domain of f(x) is (−∞,∞)−{3}
On the other hand, the Range of f(x) is −1≤f(x)≤1.
So, Domain of function = (−∞,∞)−{3}
Range of function= −1≤f(x)≤1.
Ques: Find the domain and range of the relation R: {(a,a2) | a ∈ A, a2 ∈ A} which is defined on A×A and the set A = {1,2,3,4,5,7,8,9}? (2 marks)
Ans: Relation R is defined as,
R = {(1,1), (2,4), (3,9)}
- Domain of R = {1,2,3}
- Range of R = {1,4,9}
- Codomain of R = Set A = = {1,2,3,4,5,7,8,9}
Ques: For the given function f(x) = x + 3 and g(x) = x - 1 find the value of fog(x)? (2 marks)
Ans: The given two functions are f(x) = x + 3 and g(x) = x - 1.
The function fog(x) is to be found.fog(x) = f(g(x))
= f(x-1)
= 1(x - 1) + 3
= x - 1 + 3
= x - 2
Ques: Find out the range of function: ƒ(x) = √25 – x2? (3 marks)
Ans: The root value found inside can never be negative, which is why x2 should be less than 16.
This means x ∈ [-5,5]. It is known as the domain of the function.
For range, let y = √25 – x2
Which means, y2 = 25 – x2
or x2= 25 – y2
Now, because x ∈ [-5,5].
The Range of f = [0, 5].
Ques. Express a relation R from A to A = {1, 2, 3, 4, 5} as R = {(x, y) : y = x + 1} and also find the domain, codomain and range of R? (4 marks)
Ans: As per the given question, A = {1, 2, 3, 4, 5} is the domain and codomain of R.
Now, in order to find the range, the values of y for every value of x needs to determined, that is, when x = 1, 2, 3, 4, 5. Thus:
x = 1, y = 1 + 1 = 2;
x = 2, y = 2 + 1 = 3;
x = 3, y = 3 + 1 = 4;
x = 4, y = 4 + 1 = 5;
x = 5, y = 5 + 1 = 6;
Since 6 is not part of A, thus the relation R is defined on A.
Hence, range of R = (2,3,4,5)
Thus,
Domain = Codomain = (1,2,3,4,5),
Range = (2,3,4,5)
Ques: For the given function f(x) = 2x + 3 and g(x) = x - 1 find the value of fog(x)? (2 marks)
Ans: The given two functions are f(x) = x + 3 and g(x) = x - 1.
The function fog(x) is to be found.fog(x) = f(g(x))
= f(x-1)
= 2(x - 1) + 3
= 2x - 2 + 3
= 2x - 1
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