Question

For ƒ∶(All Real Numbers) →(All Real Numbers),

Let ƒ(x) = x³

Is this an injunctive or surjective function or both (a bijection)? Show your work.

Answer 1

f:R→R, is given in the question.

Take x,y∈R, then to show injectivity, we need to show that: if f(x)=f(y) then we have x=y. We can show that x3=y3 ⟹ x=y .

Take y∈R, then to show surjectivity, we need to show that: there exists some x∈R such that y=f(x) We can show that ∀y∈R ∃x∈R: y=x3.

Bijectivity is true only when both injectivity and surjectivity are true.

Since, here both injectivity and surjectivity are true, so this is a surjective function.

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