Given the function (f: mathbb{R} rightarrow mathbb{R}) defined as (f(x) = x^2), determine if this function is surjective (onto).

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Accepted Answer

Step:1 ：Step 1: A function (f: A rightarrow B) is said to be surjective (onto) if for every (b in B), there exists an (a in A) such that (f(a) = b).Step:2 ：Step 2: In our case, (A = B = mathbb{R}), the set of all real numbers, and (f(x) = x^2).Step:3 ：Step 3: For the function to be surjective, every real number should be the square of some real number. But we know that the square of any real number is always non-negative, i.e., (x^2 geq 0) for all (x in mathbb{R}).Step:4 ：Step 4: Therefore, for any real number (b < 0), there is no real number (a) such that (a^2 = b). This means that the function (f(x) = x^2) is not surjective.

Answer

Step: 1 ：Step 1: A function (f: A rightarrow B) is said to be surjective (onto) if for every (b in B), there exists an (a in A) such that (f(a) = b).Step: 2 ：Step 2: In our case, (A = B = mathbb{R}), the set of all real numbers, and (f(x) = x^2).Step: 3 ：Step 3: For the function to be surjective, every real number should be the square of some real number. But we know that the square of any real number is always non-negative, i.e., (x^2 geq 0) for all (x in mathbb{R}).Step: 4 ：Step 4: Therefore, for any real number (b < 0), there is no real number (a) such that (a^2 = b). This means that the function (f(x) = x^2) is not surjective.

Given the function $f : R \to R$ defined as $f (x) = x^{2}$ , determine if this function is surjective (onto).

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Popular Problems

Given the function f:R→R defined as f(x)=x2, determine if this function is surjective (onto).

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Popular Problems

Given the function $f : R \to R$ defined as $f (x) = x^{2}$ , determine if this function is surjective (onto).