Question

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined by $\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{x}+\left|\mathrm{x}\right|$, then $\mathrm{f}\left(2\mathrm{x}\right)-\mathrm{f}(-\mathrm{x})-6\mathrm{x}=$

• A. f(x)
• B. 2f(x)
• C. -f(x)
• D. f(-x)

Answer 1

The correct option is A

f(x)

Step 1: Find $\mathit{f}\mathbf{\left(}\mathbf{2}\mathit{x}\mathbf{\right)}$

Given function:$\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{x}+\left|\mathrm{x}\right|$ …(1)

Put $x=2x$ in equation (1)

$\mathrm{f}\left(2\mathrm{x}\right)=6\mathrm{x}+\left|2\mathrm{x}\right|$

Step 2: Finding $f(-x)$

Put $\mathrm{x}=-\mathrm{x}$ in equation (1)

$\mathrm{f}(-\mathrm{x})=-3\mathrm{x}+\left|\mathrm{x}\right|$

Step 3: Finding $\mathrm{f}\left(2\mathrm{x}\right)-\mathrm{f}(-\mathrm{x})-6\mathrm{x}$

$\begin{array}{rcl}\therefore \mathrm{f}\left(2\mathrm{x}\right)-\mathrm{f}(-\mathrm{x})-6\mathrm{x}& =& 6\mathrm{x}+\left|2\mathrm{x}\right|-(-3\mathrm{x}+\left|\mathrm{x}\right|)-6\mathrm{x}\\ & =& 6\mathrm{x}+2\left|\mathrm{x}\right|+3\mathrm{x}-\left|\mathrm{x}\right|-6\mathrm{x}\\ & =& 3\mathrm{x}+\left|\mathrm{x}\right|\\ & =& \mathrm{f}\left(\mathrm{x}\right)\end{array}$

Hence, the correct option is an option (A).