Answer

Verified

373.5k+ views

**Hint:**To find the domain and range of the given function, we have to form a general equation for $ x $ and $ y $ . Then we need to check whether every element in $ x $ has its image or not. And we need to find in what category the values of $ x $ and $ y $ come under.

**Complete step-by-step solution:**

Let us consider the given equation,

$ f(x) = y = \dfrac{1}{{1 + x}} $

$ x,y \in \mathbb{R} $ , for any value of $ x $ , we have an image in $ y $ , except when $ x = - 1 $ .

To find the general equation for $ x $ , we solve the above equation and we get,

$

\Rightarrow y(1 + x) = 1 \\

\Rightarrow y + xy = 1 \\

\Rightarrow xy = 1 - y \\

\Rightarrow x = \dfrac{{1 - y}}{y} \\

$

$ x,y \in \mathbb{R} $ , for any value of $ y $ , we have pre image $ x $ , except $ y = 0 $

This is the required equation for $ x $ which is the preimage of $ y $ . For $ x = - 1 $ , we don’t have an image in $ y $ because when we substitute $ x = - 1 $ in $ y $ we get, $ \dfrac{1}{0} $ which is undefined.

And also, for the image $ y = 0 $ , the value of $ x $ will also be undefined. And hence the domain and range of the function will be the real numbers.

**Additional information:**There are different types of function they are one-one function, into function, onto function and bijective function. These types define the nature of the function with the help of domain, range and its co-domain.

**Note:**Let us consider a function $ f(x) = y = {x^2} $ , when we put $ x = 1 $ , we get $ y = 1 $ . Here the value of $ x $ is considered as a domain and the value $ y = 1 $ is considered as a range of the domain $ x = 1 $ . If any of the domain $ x $ is present without the image in $ y $ , then it is not a function.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Write the 6 fundamental rights of India and explain in detail