**Determining the domain and range of a function :**

Before we are trying to determine the domain and range of a function, first let us come to know, what is domain.

Let y = f(x) be a function.

Domain is nothing but the values of "x" for which the given function is defined.

Range is nothing but the values of "y" that we get for the the given domain (values of "x").

If we want to determine the domain of a particular function, we have to find the values of "x" for which the function is clearly defined.

For example,

In the function f(x) = √(x-3), f(x) is defined when x ≥ 3.

When we plug x = 3 in f(x), its value becomes zero.

When we plug x < 3 in f(x), we will have negative sign inside the radical sign and the value of f(x) becomes imaginary.

So, f(x) is defined for x ≥ 3

Hence the domain of f(x) = √(x-3), is x ≥ 3.

That is, Domain = [3, -∞)

And the range is all values that we get for f(x) when plug the values of "x" which is greater than or equal to 3.

To understand, how to determine the domain and range of rational functions, let us consider the example given below.

Let **y = 1/ (x-2)**

**Domain :**

In the above function, if we plug x = 2, the denominator will become zero and f(x) becomes undefined.

So, f(x) is defined for all real values of "x" except x = 2.

Hence, **the domain = R - {3} **

**Range :**

To find range of the rational function above, first we have to find inverse of "y".

To find inverse of "y", follow the steps given below.

**Step 1:**

y = 1/(x-2) has been defined by "y" in terms "x".

The same function has to be redefined by "x" in terms of "y".

**Step 2:**

y = 1/(x-2) ===> y(x-2) = 1

===> xy - 2y = 1

===> xy = 2y + 1

===> x = (2y + 1)/y

Now the function has been defined by "x" in terms of "y".

**Step 3:**

In x = (2y+1)**/**y, we have to replace **"x"** by **y⁻¹** and **"y"** by **"x"**

Then we will get, **y⁻¹ = (2x+1)/x**

**Step 4:**

Now, find the domain of **y⁻¹.**

In the inverse function y⁻¹, if we plug **x = 0**, the denominator becomes zero. So **y⁻¹** is undefined.

Hence, y⁻¹ is defined for all real values of "x" except x = 0.

Hence, **Domain ( y⁻¹) = R - {0}**

And we already know the fact that

**Range (y) = ****Domain (y⁻¹)**

Therefore **Range (y) = R - {0}**

To know more about determining domain and range of other functions, please click here.

Domain and range of rational functions with holes

Domain and range of rational function graphs

Domain and range of logarithmic functions

Domain and range of trigonometric functions

Domain and range of inverse trigonometric functions

After having gone through the stuff given above, we hope that the students would have understood "Determining the domain and range of a function".

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