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Q.
The domain and range of the real function $f$ defined by $f(x)=|x-1|$ is
Relations and Functions
Solution:
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{cases}
x-1, & \text{if $x \ge\,1$} \\[2ex]
-(x-1), & \text{if $x <\,1$ }
\end{cases}$
$\Rightarrow $ $f(x) =
\begin{cases}
x-1, & \text{if $x \ge\,1$} \\[2ex]
1-x, & \text{if $x <\,1$ }
\end{cases}$
From above, we observe that in both cases $f \left(x\right) \ge\, 0$.
Hence, range of $f(x)$ is $[0, \infty).$