Question

Let f : Z → Z be defined by f(a) = 2a 2 − a and g : Z → Z be defined by g(x) = x(2x − 1). Determine whether f is equal to g. Justify your answer.

Answer 1

We say two functions $f$ and $g$ are equal if they have the same domain and the same codomain, and if for every $t$ in the domain, $f(t)=g(t).$

Let $f : \Z → \Z$ be defined by $f(a) = 2a^ 2 − a.$

Domain: $\Z$

Codomain: $\Z$

$\forall t\in\Z:$ $f(t)=2t^2-t.$

Let $g : \Z → \Z$ be defined by $g(x) = x(2x-1)$

Domain: $\Z$

Codomain: $\Z$

$\forall t\in\Z:$ $g(t)=t(2t-1)=2t^2-t=f(t).$

Therefore $f$ is equal to $g$ on $\Z.$