‹ Back to the paper
Answer the following.
Prove that the function f: mathbbZ → mathbbZ, defined by f(x) = x - 5 is a bijective function.
Prove that the function $f: \mathbb{Z} \to \mathbb{Z}$, defined by $f(x) = x - 5$ is a bijective function.
Given: $f: \mathbb{Z} \to \mathbb{Z}$, defined by $f(x) = x - 5$
To show: $f$ is a bijective function.
Answer
Answer
$f(x)=x-5$ is one-one and onto, hence bijective.
- 1. One-one: let $x_1,x_2\in\mathbb{Z}$ with $f(x_1)=f(x_2)$.
- 2. Then $x_1-5=x_2-5$ (definition of $f$), so $x_1=x_2$. Hence $f$ is one-one (injective).
- 3. Onto: let $y\in\mathbb{Z}$ be any element of the codomain. Take $x=y+5$.
- 4. Since $y\in\mathbb{Z}$, $x=y+5\in\mathbb{Z}$, and $f(x)=(y+5)-5=y$. So every $y$ has a pre-image: $f$ is onto (surjective).
- 5. $f$ is both one-one and onto, so $f$ is a bijective function.