Question 11
If f (x) = 2x and g (x) = 𝑥^2/2+1 , then which of the following can be a discontinuous function
(A) 𝑓 (𝑥) + 𝑔 (𝑥) (B) f (x) – g (x)
(C) 𝑓 (𝑥) . 𝑔 (𝑥) (D) 𝑔(𝑥)/(𝑓(𝑥))
Given functions
𝑓(𝑥)=2𝑥 & 𝑔(𝑥)=𝑥^2/2+1
Checking each option one by one
Option 1
𝒇 (𝒙)+ 𝒈 (𝒙)=2𝑥+𝑥^2/2+1
=𝟏/𝟐 𝒙^𝟐+𝟐𝒙+𝟏
Continuous since it is a polynomial function
Option 2
𝒇 (𝒙)− 𝒈 (𝒙)=2𝑥−(𝑥^2/2+1)
=2𝑥−𝑥^2/2−1
=−𝟏/𝟐 𝒙^𝟐+𝟐𝒙−𝟏
Continuous since it is a polynomial function
Option 3
𝒇 (𝒙)∙𝒈 (𝒙)=2𝑥∙(𝑥^2/2+1)
=(2𝑥^3)/2+2𝑥
=𝒙^𝟑+𝟐𝒙
Continuous since it is a polynomial function
Option 4
(𝒈(𝒙))/(𝒇(𝒙))=((𝑥^2/2+ 1))/2𝑥
=(((𝑥^2 + 2)/2))/2𝑥
=(𝒙^𝟐+𝟐)/𝟒𝒙
Not defined when 4𝑥=0
i.e., 𝒙=𝟎
∴ (𝑔(𝑥))/(𝑓(𝑥)) is a discontinuous function when 𝑥=0
So, the correct answer is (D)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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