Let $f : \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x) = x^2$. Determine whether $f$ is one-one and whether it is onto.

Let f : mathbbZ → mathbbZ be defined by f(x) = x². Determine whether f is one-one and whether it is onto.

Exercise: One-One and Onto Functions · Q19

f : \mathbb{Z} \to \mathbb{Z}

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Not one-one: $f(1)=1$ and $f(-1)=1$ , so $f(1)=f(-1)$ but $1\neq-1$ . Not one-one.

Not onto: the range of $f$ is $\{0,1,4,9,16,\dots\}$ , the perfect squares (all $\geq0$ ). But the codomain is all of $\mathbb{Z}$ , which includes negative integers. Take $y=-1$ : no integer $x$ satisfies $x^2=-1$ . …

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Q.Let $f : \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x) = x^2$ . Determine whether $f$ is one-one and whether it is onto.

Concept understanding — Onto (Surjective) Functions and Bijections

Q.Let f:Z→Zf : \mathbb{Z} \to \mathbb{Z}f:Z→Z be defined by f(x)=x2f(x) = x^2f(x)=x2. Determine whether fff is one-one and whether it is onto.

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Q.Let $f : \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x) = x^2$ . Determine whether $f$ is one-one and whether it is onto.

Unlock everything free for 14 days