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Lec 14

The document contains a series of mathematical questions and answers related to functions, including definitions and properties of various functions. Each question presents a function or a set of functions, followed by multiple-choice answers. The document concludes with the correct answers to the questions, indicating the correct options for each query.

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15 views12 pages

Lec 14

The document contains a series of mathematical questions and answers related to functions, including definitions and properties of various functions. Each question presents a function or a set of functions, followed by multiple-choice answers. The document concludes with the correct answers to the questions, indicating the correct options for each query.

Uploaded by

amitbanik2007.ab
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Function

(Lecture-14)
JEE(Main+Advanced)

SUBSCRIBE

PRINCE GALLERY
JEE ADVANCED
Q1. 𝒇 ∶ 𝑹 → 𝑹 such that 𝒇 𝒙 = 𝒍𝒏 𝒙 + 𝒙𝟐 + 𝟏 . Another function 𝒈(𝒙) is defined such
that 𝒈𝒐𝒇 𝒙 = 𝒙 ∀ 𝒙 ∈ 𝑹. Then 𝒈(𝟐) is -
𝒆𝟐 +𝒆−𝟐 𝒆𝟐 −𝒆−𝟐
(A) (B) 𝒆𝟐 (C) (D)𝒆−𝟐
𝟐 𝟐
𝒙𝟐 − 𝟑𝒙 + 𝟒 ; 𝒙 < 𝟑 𝒙+𝟔 ; 𝒙<𝟒
Q2. Let 𝒇 𝒙 = ቊ and 𝒈 𝒙 = ቊ 𝟐 , then which of the
𝒙+𝟕 ; 𝒙≥𝟑 𝒙 + 𝒙 + 𝟐 ; 𝒙 ≥ 𝟒
following is/are true -
(A) 𝒇 + 𝒈 𝟏 = 𝟗 (B) 𝒇 − 𝒈 𝟑. 𝟓 = 𝟏
𝒇 𝟖
(C) 𝒇 𝒈 𝟎 = 𝟐𝟒 (D) 𝟓 =
𝒈 𝟑
Paragraph for Question 3 & 4
𝟐
𝒙 ; 𝒙<𝟎 𝒙 ; 𝒙 < −𝟏
Let 𝒇 𝒙 = ቊ and 𝒈 𝒙 = ቐ𝟐𝒙 + 𝟑 ; −𝟏 ≤ 𝒙 ≤ 𝟏
𝟏−𝒙 ; 𝒙≥𝟎
𝒙 ; 𝒙>𝟏
On the basis of above information, answer the following questions :
Q3. Range of 𝒇(𝒙) is -
(A) (−∞, 𝟏] (B) (−∞, ∞) (C) (−∞, 𝟎] (D) (−∞, 𝟐]
Q4. Range of 𝐠 𝒇 𝒙 is -
(A) (−∞, ∞) (B) 𝟏, 𝟑 ∪ (𝟑, ∞) (C) [𝟏, ∞) (D) 𝟎, ∞
Q5. If 𝒇 𝒙, 𝒚 = 𝒎𝒂𝒙 𝒙, 𝒚 + 𝒎𝒊𝒏 𝒙, 𝒚 𝒂𝒏𝒅 𝒈 𝒙, 𝒚 = 𝒎𝒂𝒙 𝒙, 𝒚 − 𝒎𝒊𝒏 𝒙, 𝒚 , then the value
𝟐 𝟑
of 𝒇 𝒈 − , − , 𝒈 −𝟑, −𝟒 is greater than –
𝟑 𝟐
(A) 1 (B) 2 (C) 3 (D) 4
Q6. If functions 𝒇 𝒙 𝒂𝒏𝒅 𝒈 𝒙 are defined on 𝑹 → 𝑹 such that
𝒙 + 𝟑 , 𝒙 ∈ 𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍
𝒇 𝒙 =ቊ , 𝐠 𝒙 = ቊ𝒙 + 𝟓 , 𝒙 ∈ 𝒊𝒓𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 then (𝒇 − 𝒈)(𝒙) is –
𝟒𝒙 , 𝒙 ∈ 𝒊𝒓𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 −𝒙 , 𝒙 ∈ 𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍
(A) one-one and onto (B) neither one-one nor onto
(C) one-one but not onto (D) onto but not one-one
Q7. Let 𝒇: 𝑨 → 𝑩 be an onto function such that
𝒇 𝒙 = 𝒙 − 𝟐 − 𝟐 𝒙 − 𝟑 − 𝒙 − 𝟐 + 𝟐 𝒙 − 𝟑 , then set ‘𝑩’ is –
(A) −𝟐, 𝟎 (B) 𝟎, 𝟐 (C) −𝟑, 𝟎 (D) −𝟏, 𝟎
𝟏 𝒇𝟒 𝒙
𝒇𝟐 𝒙 𝒇𝟑 𝒙
Q8. If 𝒇𝟏 𝒙 = 𝟐 , where 𝒇𝟐 𝒙 = 𝟐𝟎𝟏𝟐 . Where 𝒇𝟑 𝒙 = ,
𝟐𝟎𝟏𝟑
where 𝒇𝟒 𝒙 = log 𝟐𝟎𝟏𝟑 log 𝒙 𝟐𝟎𝟏𝟐, then the range of 𝒇𝟏 (𝒙) is -
(A) (𝟐, ∞) (B) (𝟐𝟎𝟏𝟐, ∞) (C) (𝟎, ∞) (D) (−∞, ∞)
𝒙𝟐 ; 𝟎<𝒙<𝟐
Q9. Let 𝒇 𝒙 = ቐ 𝟐𝒙 − 𝟑 ; 𝟐 ≤ 𝒙 < 𝟑 Then :-
𝒙+𝟐 ; 𝒙≥𝟑
𝟑 𝟑 𝟓 𝟓
(A) 𝒇 𝒇 𝒇 =𝒇 (B) 𝟏 + 𝒇 𝒇 𝒇 =𝒇
𝟐 𝟐 𝟐 𝟐
(C) 𝒇 𝒇 𝟏 =𝒇 𝟏 =𝟏 (D) none of these
Q10. Which of the following statement(s) is(are) correct ?
(A) If 𝒇 is a one-one mapping from set A to A, then 𝒇 is onto
(B) If 𝒇 is an onto mapping from set A to A, then 𝒇 is one-one
(C) Let 𝐟 𝐚𝐧𝐝 𝒈 be two functions defined from 𝑹 → 𝑹 such that 𝒈𝒐𝒇 is injective, then
𝒇 must be injective.
(D) If set A contains 3 elements while set B contains 2 elements, then total number
of functions from A to B is 8.
Q. Answer
1 C
2 ABC
3 A
4 C
5 A
6 B

7 A

8 A

9 ABC

10 CD
• Question discussion

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