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Piecewise Function

Domain and Range of a Function

Last Updated : 06 Feb, 2025

In mathematics, a function represents a relationship between a set of inputs and their corresponding outputs. Functions are fundamental in various fields, from algebra to calculus and beyond, as they help model relationships and solve real-world problems.

A function represents a relationship between a set of inputs (domain) and their corresponding outputs (range), where each input has exactly one output.

Domain: The set of all possible input values for which the function is defined.
Range: The set of all possible output values produced by the function when the input values from the domain are plugged in.
Co-domain: The co-domain of a function is the set of all possible output values that the function could potentially produce.

For example, For the given function f(x) = x³.

Domain-and-Range

Given the function f(x) = x³, the set of ordered pair is:
f(x) = {(1, 1), (2, 8), (3, 27), (4, 64)}
Domain = {1, 2, 3, 4}
Co-domain = {1, 2, 3, 4, 8, 9, 16, 23, 27, 64}
Range = {1, 8, 27, 64}

Table of Content

Domain and Range of Trigonometric & Inverse Trigonometric Functions

Domain of a Function

The domain of a function is the set of all possible input values (usually x) that the function can accept without causing any issues, such as division by zero or taking the square root of a negative number.

Example:

Function f(x) = 1/x,
- Domain: All real numbers except x = 0(division by zero is undefined).
Function g(x) = √x,
- Domain: All non-negative real numbers (x ≥ 0), because you cannot take the square root of a negative number in the set of real numbers.

Rules for Finding Domain of a Function

Check the Function's Expression: Start by analyzing the function’s formula to identify the type of function (e.g., quadratic, square root, rational, etc.)

Domain of the Polynomial functions (linear, quadratic, cubic, etc) function is R (all real numbers).
Domain of the square root function √x is x ≥ 0.
Domain of the exponential function is R.
Domain of the logarithmic function is x > 0.
We know that, the domain of a rational function y = f(x), denominator ≠ 0.

How to Find the Domain of a Function?

To find the domain of a function, use the following steps:

Step 1: First, check whether the given function can include all real numbers.
Step 2: Then check whether the given function has a non-zero value in the denominator of the fraction and a non-negative real number under the denominator of the fraction.
Step 3: In some cases, the domain of a function is subjected to certain restrictions, i.e., these restrictions are the values where the given function cannot be defined. For example, the domain of a function f(x) = 2x + 1 is the set of all real numbers (R), but the domain of the function f(x) = 1/ (2x + 1) is the set of all real numbers except -1/2.
Step 4: Sometimes, the interval at which the function is defined is mentioned along with the function. For example, f (x) = 2x² + 3, -5 < x < 5. Here, the input values of x are between -5 and 5. As a result, the domain of f(x) is (-5, 5).

After taking all the steps discussed above the set of numbers left with us is considered the domain of a function.

Solved Example: Find the domain of f(x) = 1/(x² - 1)

Solution:

Given,
f(x) = 1/(x² - 1)
Now, putting x = -1, 1 in f(x)
f(-1) = 1/{(-1)² - 1} = 1/0 = ∞
f(1) = 1/{(1)² - 1} = 1/0 = ∞
Thus, on -1 and 1 the function is f(x) is undefined and apart form that at all points the f(x) is defined. Thus, the domain of f(x) is R - {-1, 1}

Range of a Function

The Range of a Function is the set of all possible output values (usually y) that the function can produce when you plug in the values from the domain.

Example:

Function f(x) = 1/x,
- Range: All real numbers except y = 0(the output 1/x never equals zero).
Function g(x) = √x,
- Domain: All non-negative real numbers (y ≥ 0), because the square root of a non-negative number is also non-negative.

Rules of Finding Range of a Function

Check the Function’s Type: Identify the type of function you’re working with (e.g., quadratic, square root, rational, trigonometric, etc.)

For linear function the range is R.
For quadratic function y = a(x - h)² + k the range is:
- y ≥ k, if a > 0
- y ≤ k, if a < 0
For the square root function, the range is y ≥ 0.
For the exponential function, the range is y > 0.
For the logarithmic function, the range is R.

How to Find the Range of a Function?

The range or image of a function is a subset of a co-domain and is the set of images of the elements in the domain.

To find the Range of a Function use the following steps

Let us consider a function y = f(x).
Step 1: Write the given function in its general representation form, i.e., y = f(x).
Step 2: Solve it for x and write the obtained function in the form of x = g(y).
Step 3: Now, the domain of the function x = g(y) will be the range of the function y = f(x).

Thus, the range of a function is calculated.

Soved Example: Find the range of the function f(x) = 1/ (4x − 3).

Solution:

Given, f(x) = 1/ (4x − 3)
Let the function be f(x) = y = 1/ (4x − 3)
y(4x − 3) = 1
4xy - 3y = 1
4xy = 1 + 3y
x = 4y / (1 + 3y)
Here, we observe that x is defined for all the values except of y for y = −1/3 as on y = -1/3, we get an undefined value of x.
So, the range of f(x) = 1/ (4x − 3) is (−∞, −1/3) U (1/3, ∞

Interval Notation of Domain and Range

The domain and range of a function can be written in Interval Notation. For example, for f(x) = sin⁡(x):

Domain: (−∞,+∞)
Range: [−1, 1]

We use (), [], and { } to represent the domain and range of a function.

Co-Domain and Range

Co-domain is the set of the values including the range of the function nd it can have some additional values. Range is the Subset of the Codomain. This is explained using the example,

Given function, f(x) = cos x, such that, f:R→R, then

Codomain of f(x) = R
Range of R = (-1, 1)

How to Find Domain and Range

Now to calculate the domain and range of any given function study the following example carefully:

Given:

X = {1, 2, 3, 4, 5}
Y = {1, 2, 4, 5, ..., 45, 46, 47, 48, 49, 50}
Function: f(x) = x²

Calculating Domain and Range of a Function

Domain: The domain of the function consists of all the input values that the function can accept. In this case, the domain is the set X, which is:

Domain = {1, 2, 3, 4, 5}

Range: The range consists of the set of output values produced by the function. For f(x) = x², we calculate the outputs for each element in the domain:

f(1) = 1² = 1
f(2) = 2² = 4
f(3) = 3²= 9
f(4) = 4²= 16
f(5) = 5² = 25

So, the range is:

Range of F(x) = {2, 3, 4, 5, 6}

Explanation: The domain of a function is the set of possible input values. The range is the set of possible output values the function can produce based on the domain. In this example, the range of f(x) is a subset of Y, but it does not cover all values in Y.

For example, if we are given a function F: X → Y, such that F(x) = y + 1, and X = {1, 2, 3, 4, 5} and Y = {1, 2, 3, 4, 5, 6}. Here,

Domain of F(x) = X = {1, 2, 3, 4, 5}
Range of F(x) = {2, 3, 4, 5, 6}

Y is the codomain of F(x) but not the range.

Domain and range of various types of functions are discussed in the next sections.

Domain and Range of Functions: Examples

Linear Functions

A linear function is a polynomial function of degree 1, represented as f(x) = mx + b, where mmm is the slope and b is the y-intercept.

Example: For f(x) = 2x + 3, the domain and range are all real numbers. There are no restrictions on x or f(x), meaning any real number can be used as input, and any real number will be the output.
Domain: ℝ (All real numbers)
Range: ℝ (All real numbers)

Quadratic Functions

A quadratic function is a polynomial function with degree 2 i.e. f(x): ax² + bx = c = 0 is a Quadratic Function.

Example: For g(x) = x²− 4, the domain is all real numbers, but the range is restricted. Since the parabola opens upwards, the output cannot be less than -4.
Domain: ℝ (All real numbers)
Range: y ≥ −4

Read More: How to Find Range of Quadratic Function

Rational Functions

A rational function is a function expressed as the ratio of two polynomials, represented as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

Example: For h(x) = 1/(x - 2) , the domain excludes x = 2since division by zero is undefined. The range is all real numbers except 0, because the function never outputs 0.
Domain: x ≠ 2
Range: y ≠ 0

Exponential Functions

An exponential function is a function where the variable appears in the exponent, represented as f(x) = a^x, where a is a positive constant base.

Example: f(x) = a^x (where a > 0) has the following domain and range:
Domain: ℝ (all real numbers)
Range: y > 0 (the set of all positive real numbers)

Domain and Range of Trigonometric & Inverse Trigonometric Functions

For trigonometric functions, the domain is a set of all real numbers (except some values in some functions) and the range of the trigonometric functions varies with different trigonometric functions

Read More: Domain and Range of Trigonometric & Inverse Functions

Absolute Value Function

Absolute functions also called modulus function are the functions that are defined for all real numbers but their output is only positive real numbers, an absolute function only gives a positive output.

Example: An absolute value function, defined as f(x) = ∣ax + b∣, always produces non-negative outputs.
Domain: ℝ (all real numbers)
Range: [0, ∞) (all non-negative real numbers)

Square Root Function

For a square root function, the domain and range are calculated as:

Example: Suppose the square root function is, f(x) = √(ax + b)
We know that the square root of a negative number is not defined, so the domain of the square root function is,
• Domain = x ≥ -b/a = [-b/a,∞)
Now for the range of the square root function, we know that an absolute square root only gives positive values so the range is all positive real numbers.
• Range = R⁺

Logarithmic Function

Log function or the Logarithmic function are the function of the form, y = ln x and the domain nd range of the log function is:

Domain of Log function: (0, ∞)
Range of Log function: (-∞, +∞)

Greatest Integer Function

Greatest Integer Function is also called the step function and is the function that give the output as nearest integer less than or equal to the given number.

Domain of Greatest Interger Funcion: R
Range of Greatest Interger Funcion: Z

Domain and Range of a Function Graph

If the graph of any function is given then finding the domain and range is very easy task. Suppose we are given any curve then finding wether the curve is function or not is our first priority and this is found using the vertical line test. Then if the curve is given in the form y = f(x), then the projection on the graph on the x-axis gives the Domain of the function and the projection of the graph on the y-axis gives the range of the function.

Domain and Range Worksheets

Download Domain and Range Worksheet 1	Download Domain and Range Worksheet 2

Domain and Range of Common Functions

Here is the list of domain and range of common functions:

Function Type	Example	Domain	Range
Linear Function	f(x) = 2x + 3	All real numbers (ℝ)	All real numbers (ℝ)
Quadratic Function	f(x) = x² - 4	All real numbers (ℝ)	y ≥ -4 ( if a > 0)
Square Function	f(x) = √x	x ≥ 0	y ≥ 0
Exponential Function	f(x) = e^x	All real numbers (ℝ)	y > 0
Logarithmic Function	f(x) = ln(x)	x > 0	All real numbers (ℝ)
Rational Function	f(x) = 1/x	All real numbers except x = 0	All real numbers except y = 0

Domain and Range Practice Problems

Practice problems on Domain and Range of a function are provided below:

Question 1: Find the domain of a function f(x) = (2x + 1)/ (x² − 4x + 3).

Solution:

Given Function:
f(x) = (2x + 1)/ (x² − 4x + 3)
f(x) = (2x + 1)/ (x − 1)(x − 3)
Observing the function we can say that the function f(x) is defined for all the values of x except for the values where, the denominator of the function is zero.
So f(x) is not defined when, (x − 1)(x − 3) = 0
This can be acheived if of the bracket is zero, i.e.
x − 1 = 0 => x = 1 is where the function f(X) is undefined.
x − 3 = 0 => x = 3 is where the function f(X) is undefined.
Thus, the domian of f(x) is all the values except {1, 3}
Domain of f(x) = R − {1, 3}
Hence, the domain of the given function f(x) is R − {1, 3}.

Question 2: Find the domain and range of a function f(x) = x² + 1.

Solution:

Given Function:
f(x) = x² + 1
This is a polynomial function and we know that a polynomial function is defined for all the values of x.
Thus, f(x) is defined for all x
Domain of f(x) = R = (-∞, ∞)
For Range,
Let f(x) = y = x² + 1
y = x² + 1
⇒ x² = y − 1
⇒ x = √(y − 1)
The square root of the function is defined for all the vaues except for the negative values.
So, (y − 1) ≥ 0
y ≥ 1
Thus, the range of the function is, [1, ∞)

Question 3: Find the domain and range of a function f(x) = (x + 2)/ (x – 3).

Solution:

Given Function, f(x) = (x + 2)/ (x – 3)
Observing the function we can say that the function f(x) is defined for all the values of x except for the values where, the denominator of the function is zero.
Thus the function is defined for all the values of x but not where x - 3 = 0
x - 3 = 0
⇒ x = 3
So, the domain of f(x) is R - {3}
For Range,
Let y = f(x)
⇒ y = (x + 2)/ (x – 3)
⇒ y(x - 3) = (x + 2)
⇒ xy – 3y = x + 2
⇒ xy – x = 3y + 2
⇒ x (y – 1) = 3y + 2
⇒ x = (3y + 2)/ (y – 1)
Observing the above equation we can say that x is defined for all the values except for the values where the denominator of the functiuon is zero, i.e.
y - 1 = 0
⇒ y = 1
Range of f(x) = R - {1}

Question 4: Find the domain and range of a function f(x) = 3e^x/7.

Solution:

Given Function,
f(x) = 3e^x/7
It is an exponential function which is defined for all the values of x.
So, Domain of f(x) is R
For Range
Let f(x) = y
y = 3e^x/7
⇒ e^x = 7y/3
⇒ x = log_e(7y/3)
We know that logarithmic functions are defined only for the positive values of x.
So, x is defined only when y > 0
Thus, range of f(x) is (0, ∞)

Articles related to Domain and Range of a Function:
Trigonometric Function Graph	Relation and Function
Range of Function	Domain and Range of a Relation

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