Question

The domain and range of the real function $f$ defined by $f(x)=|x-1|$ is

• A. $R,[0,\infty )$
• B. $R,(-\infty ,0)$
• C. $R,R$
• D. $\left(-\infty ,0\right),R$

Answer 1

Answer: (A)

We have $f\left(x\right)=\left|x-1\right|$
Here, $f\left(x\right)$ is a modulus function and since modulus of a real number is uniquely defined $\forall$ real positive number.
$\therefore$ The domain of $f\left(x\right)$ is $R$.
We see that $f\left(x\right)=\left|x-1\right|$
$f(x)=\{\begin{array}{ll}x-1,& \text{if x≥1}\\ -(x-1),& \text{if x<1 }\end{array}$
$\Rightarrow$ $f(x)=\{\begin{array}{ll}x-1,& \text{if x≥1}\\ 1-x,& \text{if x<1 }\end{array}$
From above, we observe that in both cases $f\left(x\right)\ge 0$.
Hence, range of $f(x)$ is $[0,\infty ).$