f:Q-Q, defined by f(x)=x^3+1,injective or surjective or both ?
Answer (1)
Student Expert
23rd Apr, 2019
Hello Rashmi Singh,
f(a)=f(b):
f(a)=a3+1
f(b)=b3+1
a3+1=b3+1
a3=b3 subtract 1 from each side
(a3)1/3=(b3)1/3 take the cubic root of each side
a=b
Therefore, f is injective.
Again, following that helpful answer, solve for x, and then plug into f(x):
y=x3+1
y−1=x3
x=(y−1)13
Now, plug x into f, and check if result equals y, which would prove the surjective property.
f(x)=x3+1
y=((y−1)1/3)3+1 use y rather than f(x)
y=y−1+1
y=y
Therefore, it's also surjective.
Thanku.
1 Comment
Comments (1)
Thany you Dhameer Bhatia... The answer was helpful... Can u please give answer to my first question also?
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