The function f(x) = x3is bijection from R to R. Is it True or False?a)...
The function f(x) = x3 is one to one as no two values in domain are assigned the same value of the function and it is onto as all R of the co domain is images of elements in the domain.
The function f(x) = x3is bijection from R to R. Is it True or False?a)...
The function f(x) = x^3 is a bijection from R to R.
To determine if the function f(x) = x^3 is a bijection from R to R, we need to check if it is both injective (one-to-one) and surjective (onto).
Injective (One-to-One):
A function is injective if every element in the domain maps to a unique element in the range. In other words, if f(a) = f(b), then a = b.
Let's suppose f(a) = f(b):
a^3 = b^3
Taking the cube root of both sides, we get:
a = b
This shows that if f(a) = f(b), then a = b. Therefore, the function f(x) = x^3 is injective.
Surjective (Onto):
A function is surjective if every element in the range is mapped to by at least one element in the domain. In other words, for every y in the range, there exists an x in the domain such that f(x) = y.
Let's consider an arbitrary y in the range. We need to find an x such that f(x) = y.
y = x^3
Taking the cube root of both sides, we get:
x = ∛y
This shows that for any y in the range, we can find an x in the domain such that f(x) = y. Therefore, the function f(x) = x^3 is surjective.
Bijection:
Since the function f(x) = x^3 is both injective and surjective, it is a bijection from R to R.
Therefore, the correct answer is option 'A' - True.