Question

Let f:
RR defined as f(x) = x3 + x2 + cx + d be a
bijective function then the range of c is​

Answer 1

Step-by-step explanation:

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12th

Maths

Relations and Functions

Composition of Functions

Let f(x) = 2x - sin x and g...

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Asked on December 27, 2019 by

Nilraj Oli

Let f(x)=2x−sinx and g(x)=

3

x

, then

THIS QUESTION HAS MULTIPLE CORRECT OPTIONS

A

Range of gof is R

B

gof is one-one

C

both f and g are one-one

D

both f and g are onto

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ANSWER

Given : f(x)=2x−sinx

g(x)=3

x

For all values of x , in f(x)andg(x) each element of the domain is mapped to exactly one element of the domain, Then it is known as Bijective Function.

⇒f(x)=2x−sin(x)

⇒f

1

(x)=2−cosx which is between 1 and 3 (inclusive for all x.Therefor f(x) is strictly increasing which implies it is bijective (one-one and onto)

⇒g(x)=

3

x

=(x)

3

1

⇒g

1

(x)=

3

1

(x)

3

1

−1

=

3

1

(x)

3

−2

Since for every x there is distinct values of g(x) which implies g(x) is bijective (one - one and onto)

gof=g(x)of(x)

For f=

3

x

substitute x with g(x)=2x−sinx

gof=

3

2x−sin(x)

Since sinx function range is [-1,+1]

∴2x>sinx

⇒(2x−sinx)>0

gof=

3

2x−sin(x)

will be in the range of real numbers (R) and one-one function ,since both g(x)andf(x) are one-one function.

Therefore option A,B,C,D are correct.