Question

FIND THE DOMAIN AND RANGE OF THE FOLLOWING RATIONAL FUNCTION​ USE AN NOTATION

Answer 1

Domain of the function = R - { 1 }

Range of the function = R - { 1 }

Given : The function

$\displaystyle \sf{ f(x) = \frac{2 + x}{x + 1} }$

To find : The domain and range of the function

Solution :

Step 1 of 3 :

Write down the given function

The given function is

$\displaystyle \sf{ f(x) = \frac{2 + x}{x + 1} }$

Step 2 of 3 :

Find out the domain

$\displaystyle \sf{ f(x) = \frac{2 + x}{x + 1} }$

Clearly f(x) is not well defined where denominator of f(x) vanishes

The denominator vanishes when x + 1 = 0

Which gives x = - 1

Hence domain of the function = R - { 1 }

Step 3 of 3 :

Find the range of the function

Let y = f(x)

Then we have

$\displaystyle \sf{ y= \frac{2 + x}{x + 1} }$

$\displaystyle \sf{ \implies xy + y = 2 + x }$

$\displaystyle \sf{ \implies xy - x =2 - y }$

$\displaystyle \sf{ \implies x(y - 1)=2 - y }$

$\displaystyle \sf{ \implies x = \frac{2 - y}{y - 1} }$

Since x is real

So we have y - 1 ≠ 0

⇒ y ≠ 1

Hence range of the function = R - { 1 }

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