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Lec 13

The document covers composite functions in mathematics, detailing definitions and properties such as non-commutativity and associativity. It includes several questions and answers related to finding domains, defining functions, and examining properties of composite functions. Additionally, it discusses conditions under which functions are one-to-one, onto, or bijective.

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21 views20 pages

Lec 13

The document covers composite functions in mathematics, detailing definitions and properties such as non-commutativity and associativity. It includes several questions and answers related to finding domains, defining functions, and examining properties of composite functions. Additionally, it discusses conditions under which functions are one-to-one, onto, or bijective.

Uploaded by

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

(Lecture-13)
JEE(Main+Advanced)

SUBSCRIBE

PRINCE GALLERY
JEE ADVANCED
Composite function

𝒇 ∶ 𝑨 → 𝑩, 𝒇(𝒙)
𝒈 ∶ 𝑩 → 𝑪, 𝒈(𝒙)
𝒈𝒐𝒇 ∶ 𝑨 → 𝑪, 𝒈 𝒇 𝒙
For 𝒈 𝒇 𝒙 to be defined range of 𝒇(𝒙) is subset of domain of 𝒈(𝒙).
Q1. The function 𝒇(𝒙) is defined in [𝟎, 𝟏]. Find the domain of 𝒇(𝒕𝒂𝒏𝒙).
Q2. Suppose that 𝒈 𝒙 = 𝟏 + 𝒙 𝒂𝒏𝒅 𝒇 𝒈 𝒙 = 𝟑 + 𝟐 𝒙 + 𝒙.Then find the function 𝒇 𝒙 .
Q3. If 𝒇 𝒈 𝒙 = 𝒔𝒊𝒏𝟐 𝒙 𝒂𝒏𝒅 𝒈 𝒇 𝒙 = |𝒔𝒊𝒏 𝒙|. Then which of the following may be true ?
(A) 𝒇 𝒙 = 𝒔𝒊𝒏𝟐 𝒙 ; 𝒈 𝒙 = 𝒙
(B) 𝒇 𝒙 = 𝒙𝟐 ; 𝒈 𝒙 = 𝒔𝒊𝒏 𝒙
(C) 𝒇 𝒙 = 𝒔𝒊𝒏𝟐 𝒙 ; 𝒈 𝒙 = 𝒙
(D) None of these
Q4. (a) If 𝒇 𝒙 = 𝒙𝟐 + 𝟐𝒙 + 𝟏 𝐚𝐧𝐝 𝒈 𝒇 𝒙 = |𝒙 + 𝟏| then find 𝒈(𝒙).
(b) If 𝒇 𝒙 = 𝒙𝟐 + 𝟐𝒙 + 𝟏 𝐚𝐧𝐝 𝒇 𝒈 𝒙 = 𝒙 + 𝟏 then find 𝒈(𝒙).
𝟏
Q5. If 𝒇 𝒙 = 𝒂 − 𝒙𝒏 𝒏 , 𝒙 > 𝟎, 𝒏 ≥ 𝟐 𝒂𝒏𝒅 𝒏 ∈ 𝑵, then find 𝒇𝒐𝒇𝒐𝒇𝒐𝒇 … . . 𝒏 𝒕𝒊𝒎𝒆𝒔 𝒇(𝒙).
Q6. If 𝒇 𝒙 = 𝟏 + 𝒙 − 𝒙 𝒂𝒏𝒅 𝒈 𝒙 = 𝒔𝒈𝒏 (𝒙), then find 𝒇 𝒈 𝒙 𝒂𝒏𝒅 𝒈 𝒇 𝒙 .
(where [𝒙] represents greatest integer function)
Comment on composite function :-

(i) 𝒇𝒐𝒈 𝒙 ≠ 𝒈𝒐𝒇(𝒙) [not commutative]


(ii) 𝒇𝒐𝒈𝒐𝒉 𝒙 = 𝒇𝒐𝒈 𝒉 𝒙
= 𝒇 𝒈𝒐𝒉 𝒙 [Associative]
Comment on composite function :-

(iii) If 𝒇 𝒙 𝒂𝒏𝒅 𝒈(𝒙) are one-one then 𝒇𝒐𝒈 𝒂𝒏𝒅 𝒈𝒐𝒇 are also one-one.
(iv) If 𝒇 𝒙 𝒂𝒏𝒅 𝒈(𝒙) are onto then 𝒇𝒐𝒈 𝒂𝒏𝒅 𝒈𝒐𝒇 are also onto.
Hence, if 𝒇 𝒙 𝒂𝒏𝒅 𝒈(𝒙) are bijective then 𝒇𝒐𝒈 𝒂𝒏𝒅 𝒈𝒐𝒇 are also bijective.
Q7. Let 𝒇 𝒙 𝒂𝒏𝒅 𝒈(𝒙) be bijective functions where 𝒇: 𝒂, 𝒃, 𝒄, 𝒅 → 𝟏, 𝟐, 𝟑, 𝟒 and
𝒈 ∶ 𝟑, 𝟒, 𝟓, 𝟔 → 𝒘, 𝒙, 𝒚, 𝒛 , respectively. Then find the no. of elements in the range of
𝒈 𝒇 𝒙 .
Comment on composite function :-

(v) If 𝐠𝐨𝒇 𝒙 is one-one then, then 𝒇(𝒙) is necessarily one-one but 𝒈(𝒙) may not be
one-one.
(vi) If 𝐠𝐨𝒇 𝒙 is onto then, then 𝐠(𝒙) is necessarily onto but 𝒇(𝒙) may not be onto.
Comment on composite function :-

(v) If 𝐠𝐨𝒇 𝒙 is one-one then, then 𝒇(𝒙) is necessarily one-one but 𝒈(𝒙) may not be
one-one.
(vi) If 𝐠𝐨𝒇 𝒙 is onto then, then 𝐠(𝒙) is necessarily onto but 𝒇(𝒙) may not be onto.
Composite of non-uniform function :-
𝟏 + 𝒙, 𝟎 ≤ 𝒙 ≤ 𝟐
Q8. If 𝒇 𝒙 = ቊ , then find 𝒇 𝒇 𝒙 .
𝟑 − 𝒙, 𝟐 < 𝒙 ≤ 𝟑
Composite of non-uniform function :-
𝒙 + 𝟏, 𝒙 < 𝟎 𝒙𝟑 , 𝒙<𝟏
Q9. 𝒇 𝒙 = ቊ 𝟐 and 𝐠 𝒙 = ቊ
𝒙 , 𝒙≥𝟎 𝟐𝒙 − 𝟏, 𝒙≥𝟏
Then find 𝒇 𝒈 𝒙 .
Composite of non-uniform function :-
𝒍𝒏 𝒙, 𝟎 < 𝒙 < 𝟏 𝒙 + 𝟏, 𝒙<𝟐
Q10. 𝒇 𝒙 = ቊ 𝟐 and 𝐠 𝒙 = ቊ 𝟐
𝒙 − 𝟏, 𝒙≥𝟏 𝒙 − 𝟏, 𝒙≥𝟐
Then find 𝐠 𝒇 𝒙 .
Q. Answer
1 𝝅
𝒙 ∈ 𝒏𝝅, 𝒏𝝅 +
𝟒
2 𝒇 𝒙 = 𝟐 + 𝒙𝟐
3 C
4 (i) 𝒈 𝒙 = 𝒙
(ii) 𝒈 𝒙 = −𝟏 + 𝟏 + 𝒙 𝒐𝒓 − 𝟏 − 𝟏 + 𝒙
5 𝒙 𝒊𝒇 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏
𝒇𝒐𝒇𝒐𝒇 … . . 𝒇 𝒙 = ቐ 𝟏
𝒏
𝒂− 𝒙 𝒏 𝒊𝒇 𝒏 𝒊𝒔 𝒐𝒅𝒅

6 𝒈 𝒇 𝒙 = 𝟏 𝒂𝒏𝒅 𝒇 𝒈 𝒙 =𝟏
7 𝟐
8 𝒙 + 𝟐, 𝒙 ∈ 𝟎, 𝟏
𝒇 𝒇 𝒙 = ቐ−𝒙 + 𝟐, 𝒙 ∈ 𝟏, 𝟐
−𝒙 + 𝟒, 𝒙 ∈ 𝟐, 𝟑
Q. Answer
9 𝒙𝟑 + 𝟏, 𝒙<𝟎
𝒇 𝒈 𝒙 = ൞ 𝒙𝟔 , 𝟎≤𝒙<𝟏
𝟐𝒙 − 𝟏 𝟐 , 𝒙≥𝟏
10 𝒍𝒏 𝒙 + 𝟏, 𝟎 < 𝒙<𝟏
𝟐
𝒈 𝒇 𝒙 = 𝒙 , 𝟏≤𝒙< 𝟑
𝟐
𝒙𝟐 − 𝟏 − 𝟏, 𝒙≥ 𝟑
• Question discussion

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