Let f: R → R be defined as f(x) = x^4 . Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto
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Given, f : R → R be defined as f(x) = x⁴
Let x, y ∈ R such that f(x) = f(y)
now, x⁴ = y⁴
x = ±y
means, f(1) = f(-1) = 1 but 1 ≠ -1
therefore f is not one - one .
given co - domain of function belongs to all real numbers e.g., co-domain ∈ R
but for every value of x, f(x) is always positive.
e.g., Range ∈ [ 0, ∞ )
here co-domain ≠ range
therefore , f is not onto.
hence, f is neither one - one nor onto.
option (D) is correct.
Let x, y ∈ R such that f(x) = f(y)
now, x⁴ = y⁴
x = ±y
means, f(1) = f(-1) = 1 but 1 ≠ -1
therefore f is not one - one .
given co - domain of function belongs to all real numbers e.g., co-domain ∈ R
but for every value of x, f(x) is always positive.
e.g., Range ∈ [ 0, ∞ )
here co-domain ≠ range
therefore , f is not onto.
hence, f is neither one - one nor onto.
option (D) is correct.
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