Introduction to Domain of a Function
Domain of a function is the set of all possible values which qualify as inputs to a function. To find the domain of the function, it should be defined as the entire set of values possible for independent variables.
Example: Let the function is f(x)=x². The domain of function f(x)=x² is all real numbers.
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The range of the function is defined as all the outputs of a function or it can also be obtained after substituting the domain value in the function.
Example: In the above function f(x)=x², the range value is {1,4,9...}
Domain and Range of Trigonometric Functions
Let us consider the basic trigonometric identity:
sin²x + cos²x = 1
From the given identity, the following things we can find:
cos²x = 1 - sin²x
cosx = \[\sqrt{1-sin^{2}x}\]
We know that the cosine function is defined only for real values therefore the value inside the root is always non-negative. Therefore,
1 - sin²x ≥ 0
sin x ∈ [-1, 1]
Domain of sin (x) is all real numbers.
In a similar way, we can find the domain and range for cos x.
Hence, for the trigonometric functions f(x)= sin x and f(x)= cos x, the domain will contain the entire set of real numbers because they are defined for all the real numbers. The range of f(x) = sin x and f(x)= cos x will lie from -1 to 1 including both -1 and +1. It can be represented as
-1 ≤ sin x ≤1
-1 ≤ cos x ≤1
Now, let us discuss the domain and range of the function f(x)= tan x. We know the value of tan x = sin x / cos x. It means that tan x will be defined for all values except the values where cos x = 0, because a fraction with denominator 0 is not defined. Now, we know that the value of cos x is zero for the angles π/2, 3 π/2, 5 π/2 etc.
Therefore, cos x = 0 ∀ ∈ \[\frac{(2n+1) \pi}{2}\], where n ∈ z.
Hence, tan x is not defined for these values.
So, the domain of tan x will be R - \[\frac{(2n+1) \pi}{2}\] and the range will be set of all real numbers i.e R.
As we know sec x, cosec x and cot x are the reciprocal of function cos x, sin x and tan x respectively. Thus,
sec x = 1/cos x
cosec x = 1/sin x
cot x = 1/tan x
Therefore, these ratios will not be defined for the following function:
sec x will not be defined at the points where cos x is 0. Hence, the domain of sec x is R-(2n+1)π/2, where n∈I and the range of sec x will be R- (-1,1). Since cos x lies between -1 to1. So the value of sec x can never lie between that region.
cosec x is defined at the points where sin x value is 0. Hence, the domain of cosec x is R-nπ, where n∈I. The range value of cosec x will be R- (-1,1). Since sin x lies between -1 to 1. So the value of cosec x can never lie in the region of -1 and 1.
cot x will not be defined at the points where tan x is 0. Hence, the domain value of cot x is R-nπ, where n∈I. The range of cot x is the set of all real numbers i.e R.
Domain of Sin Inverse x
Sin inverse x is an inverse trigonometric function. If we know the range of trigonometric functions, we can find the domain of inverse trigonometric functions. The range of sin x is [-1,1].
We also know that,
Range of trigonometric function = Domain of an inverse trigonometric function
So, the domain of sin inverse x is [-1,1] or -1 ≤ x ≤ 1.
Domain and Range of a Graph
We can also find the domain and range of functions by using graphs. As we know the domain refers to the set of possible input values. The domain of a graph is the set of all the input values shown on the x-axis. The range is the set of values of all the possible outputs, that are shown on the y-axis.
Ques: Find the domain and range of the function f whose graph is given below.
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Sol: We will draw a horizontal and vertical line to visualise domain range. In the figure, we can observe that the horizontal extent of the graph is from –3 to 1. So the domain of function f is (-3,1].
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The vertical line of the graph is from 0 to –4, so the range is [-4,0].
FAQs on Domain and Range of a Function
1. What is the domain and range of a function in Mathematics?
The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. The range is the set of all possible output values (usually y-values) obtained by substituting the domain values into the function.
2. How do you find the domain of a function like f(x) = √(x-2)?
For f(x) = √(x-2), the expression under the root must be non-negative. So, set x-2 ≥ 0, which gives x ≥ 2. Therefore, the domain is [2, ∞).
3. Why do some functions have restricted domains?
Functions have restricted domains when certain input values make the function undefined. Common reasons include division by zero or taking the square root of a negative number. Always check for mathematical operations that limit which x-values are acceptable.
4. What are the domain and range of trigonometric functions like sin(x), cos(x), and tan(x)?
- sin(x) and cos(x): Domain is all real numbers (ℝ); Range is [-1, 1]
- tan(x): Domain is ℝ except x ≠ (2n+1)π/2, n∈ℤ; Range is ℝ
5. Can a function have a range that includes negative numbers?
Yes, many functions have ranges that include negative numbers as outputs. For example, f(x) = x³ has a range of all real numbers, including negatives.
6. How is the range of a function affected if we shift its graph vertically?
A vertical shift moves the entire graph up or down, which changes all output values (the range) by the shift amount. The domain remains unchanged.
7. How do you determine the domain and range from a graph?
The domain includes all x-values covered by the graph (from left to right), while the range includes all y-values the graph reaches (from bottom to top). Check the extent and boundaries in both directions on the axes.
8. What common mistakes do students make when finding the range of a function?
Typical mistakes include:
- Confusing the range with domain
- Ignoring restrictions imposed by square roots or denominators
- Not considering endpoints, especially in closed intervals
- Not checking if the function can actually attain boundary values
9. Why is it important to understand the domain and range for solving NCERT Maths problems in CBSE 2025-26?
Knowing the domain and range helps avoid calculation errors and ensures correct application of concepts in CBSE exam questions. Many marks are awarded for correctly identifying these sets, as they form the foundation for further problem-solving, graph plotting, and trigonometric/inverse function analysis.
10. What is the domain of the inverse trigonometric function sin-1(x)?
The domain of sin-1(x) is [-1, 1], as only values in this interval are possible outputs (ranges) from the standard sine function.
11. What happens if you substitute a value outside the domain into a function?
If you use a value outside the domain, the function becomes undefined for that value. For instance, f(x) = 1/(x-3) is undefined at x = 3.
12. How do you write the domain and range using interval notation?
Use interval notation to clearly specify domains and ranges. Examples: [a, b] means all values between a and b (inclusive), (a, b) means between a and b but not including them, and (−∞, ∞) indicates all real values.
13. What is the effect of taking the reciprocal of a function on its domain and range?
Taking the reciprocal of a function excludes all points where the original function equals zero from the domain, since 1/0 is undefined. The range may also change significantly, depending on the original outputs.
