Functions Question 35
Question: Mapping $ f:R\to R $ which is defined as $ f(x)=\cos x,\ x\in R $ will be
[UPSEAT 1999]
Options:
A) Neither one-one nor onto
B) One-one
C) Onto
D) One-one onto
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ x_1,,x_2\in R, $ then $ f(x_1)=\cos x_1 $ , $ f(x_2)=\cos x_2 $ , so $ f(x_1)=f(x_2) $
Þ $ \cos x_1=\cos x_2 $
Þ $ x_1=2n\pi \pm x_2 $
Þ $ x_1\ne x_2 $ , so it is not one-one. Again the value of f-image of x lies in between ?1 to 1
Þ $ f[R]={ f(x):-1\le f(x)\le 1) } $ So other numbers of co-domain (besides ?1 and 1) is not f-image. $ f[R]\in R, $ so it is also not onto. So this mapping is neither one-one nor onto.
BETA
