Answer
Domain of $f\left( x \right):\left( -\infty ,\infty \right)$ , Range of $f\left( x \right):\left[ -2,\infty \right)$
Work Step by Step
The graph is a parabola. Since $f\left( x \right)$ is a polynomial, it exists for every value of $x$ on the real axis.
Therefore, the domain of $f\left( x \right)=3{{x}^{2}}-6x+1$ is $\left( -\infty ,\infty \right)$.
Since,
$\begin{align}
& 3{{x}^{2}}-6x+1=3\left( {{x}^{2}}-2x \right)+1 \\
& =3\left( {{x}^{2}}-2x+1-1 \right)+1 \\
& =3\left( {{\left( x-1 \right)}^{2}}-1 \right)+1 \\
&
\end{align}$
Also,
$\begin{align}
& {{\left( x-1 \right)}^{2}}\ge 0 \\
& {{\left( x-1 \right)}^{2}}-1\ge -1 \\
& 3\left( {{\left( x-1 \right)}^{2}}-1 \right)\ge -3 \\
& 3\left( {{\left( x-1 \right)}^{2}}-1 \right)+1\ge -2 \\
\end{align}$
So, the range of the function is $\left[ -2,\infty \right)$.
