wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If f:RC is defined by f(x)=e2ix for x belongs to R, then f is (where C denotes the set of all complex numbers)


A

One-one

No worries! We‘ve got your back. Try BYJU‘S free classes today!
B

Onto

No worries! We‘ve got your back. Try BYJU‘S free classes today!
C

One-one and onto

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

Neither one-one nor onto

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct option is D

Neither one-one nor onto


Explanation for the correct answer:

Step 1: Given information

Given, f(x)=e2ix

This can be written as,

f(x)=cos2x+isin2x

We know that cosandsin functions are periodic.

Period of sin(Ax)=2πA

Period ofsin2x: 2π2=π

Period of cos(Ax)=2πA

Period of cos2x: 2π2=π

f(π+x)=cos2(π+x)+isin2(π+x)=cos2x+isin2x=f(x)

Thus,f(x)is also periodic with a period of π.

Since periodic functions are not one-one, f(x) is not one-one.

Step 2: Finding the magnitude of f(x):

=f(x)=ei2x=cos22x+sin22xsin2θ+cos2θ=1=1

But, in a complex plane, the magnitude of a complex number can vary from 0 to infinity.

Since, for any value ofx, the magnitude of f(x) cannot be greater than or lesser than 1 (i.e., f(x) doesn't cover the entire complex plane), we can say that f(x) is not onto.

Hence, the correct answer is an option (D).


flag
Suggest Corrections
thumbs-up
4
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Periodic Functions
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon