Domain and Range 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:

1. 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.

2. 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.

3. 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].