Is the function ( f: mathbb{R} rightarrow mathbb{R}, f(x) = x^3 ) surjective?

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Accepted Answer

Step:1 ：Step 1: To determine if a function is surjective, we need to show that for every element ( y ) in the codomain ( mathbb{R} ), there exists an element ( x ) in the domain ( mathbb{R} ) such that ( f(x) = y ).Step:2 ：Step 2: Let&#x27;s take an arbitrary ( y ) in ( mathbb{R} ). We need to find an ( x ) such that ( f(x) = y ). So we solve the equation ( x^3 = y ) for ( x ).Step:3 ：Step 3: By taking the cube root on both sides, we find that ( x = sqrt[3]{y} ).Step:4 ：Step 4: Since ( sqrt[3]{y} ) is defined for all ( y ) in ( mathbb{R} ), we can conclude that for every ( y ) in ( mathbb{R} ), there exists an ( x ) in ( mathbb{R} ) such that ( f(x) = y ).

Answer

Step: 1 ：Step 1: To determine if a function is surjective, we need to show that for every element ( y ) in the codomain ( mathbb{R} ), there exists an element ( x ) in the domain ( mathbb{R} ) such that ( f(x) = y ).Step: 2 ：Step 2: Let&#x27;s take an arbitrary ( y ) in ( mathbb{R} ). We need to find an ( x ) such that ( f(x) = y ). So we solve the equation ( x^3 = y ) for ( x ).Step: 3 ：Step 3: By taking the cube root on both sides, we find that ( x = sqrt[3]{y} ).Step: 4 ：Step 4: Since ( sqrt[3]{y} ) is defined for all ( y ) in ( mathbb{R} ), we can conclude that for every ( y ) in ( mathbb{R} ), there exists an ( x ) in ( mathbb{R} ) such that ( f(x) = y ).

Is the function $f : R \to R, f (x) = x^{3}$ surjective?

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Popular Problems

Is the function f:R→R,f(x)=x3 surjective?

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Popular Problems

Is the function $f : R \to R, f (x) = x^{3}$ surjective?