# Domain and Range of a Function

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## 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)=xÂ². The domain of function f(x)=xÂ² is all real numbers.

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 functionÂ  f(x)=xÂ², the range value is {1,4,9...}

### Domain and Range of Trigonometric Functions

Let us consider the basic trigonometric identity:

sinÂ²x + cosÂ²x = 1

From the given identity, the following things we can find:

cosÂ²x = 1 - sinÂ²xÂ

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 - sinÂ²x â‰¥ 0

sin x âˆˆ [-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 â‰¤ sin x â‰¤1

• -1 â‰¤ cos x â‰¤1

Now, let us discuss the domain and range of the function f(x)= tan x. We know the value ofÂ  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 anglesÂ  Ï€/2, 3 Ï€/2, 5 Ï€/2 etc.

Therefore, cos x = 0 âˆ€ âˆˆ $\frac{(2n+1) \pi}{2}$, where n âˆˆ z.Â

Hence, tan x is not defined for these values.

So, the domain of tan xÂ  will be R - $\frac{(2n+1) \pi}{2}$ and the range will be set of all real numbers i.eÂ  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)Ï€/2, where nâˆˆI 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-nÏ€, where nâˆˆI. 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-nÏ€, where nâˆˆI. The range of cot x is the set of all real numbers i.e R.

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### 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,Â

Range of trigonometric function = Domain of an inverse trigonometric function

So, the domain of sin inverse x is [-1,1] or -1 â‰¤ x â‰¤ 1.Â

### 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.

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 â€“3 to 1. So the domain of function f is (-3,1].