Answer
Domain: $\mathbb{R}$
Range: $\{1\}\cup (\mathbb{R}\setminus \mathbb{Q})$
Work Step by Step
Find the domain:
$D_f=\{x|\text{x is rational or x is irrational}\}$
$D_f=\{x|x\in \mathbb{Q}\cup (\mathbb{R}\setminus\mathbb{Q})\}$
$D_f=\{x|x\in \mathbb{R}\}$
$D_f=\mathbb{R}$
Find the range:
$R_f=\{y=f(x)|x\in D_f\}$
$R_f=\{y|y=1\text{ if x is rational or } y=5x\text{ if x is irrational}\}$
$R_f=\{y|y=1\text{ or } y=5x\text{ is irrational}\}$
$R_f=\{1\}\cup (\mathbb{R}\setminus\mathbb{Q})$
