Question -
Answer -
Givenf: R → R is a function defined by f(x) = 4x3 +7
Injectivity:
Let x and y beany two elements in the domain (R), such that f(x) = f(y)
⇒ 4x3 +7 = 4y3 + 7
⇒ 4x3 = 4y3
⇒ x3 = y3
⇒ x = y
So, f isone-one.
Surjectivity:
Let y be any element in the co-domain (R), suchthat f(x) = y for some element x in R (domain)
f(x) = y
⇒ 4x3 +7 = y
⇒ 4x3 = y −7
⇒ x3 =(y – 7)/4
⇒ x = ∛(y-7)/4 in R
So, for every elementin the co-domain, there exists some pre-image in the domain. f is onto.
Since, f is both one-to-one and onto, it is a bijection.