Step 1

When x is less than 0 then also y is less than zero

if \(\displaystyle{x}_{{1}}\ne{x}_{{2}}{f}{\quad\text{or}\quad}{x}_{{1}},{x}_{{2}}{<}{0}\)</span>

Then, \(\displaystyle{y}_{{1}}\ne{y}_{{2}}\)

if \(\displaystyle{x}\ge{0}\) , then y is also greater than 0.

if \(\displaystyle{x}_{{3}}\ne{x}_{{4}}{f}{\quad\text{or}\quad}{x}_{{3}},{x}_{{4}}{<}{0}\)</span>

Then \(\displaystyle{2}{x}_{{3}}\ne{2}{x}_{{4}}\)

So, \(\displaystyle{y}_{{3}}\ne{y}_{{4}}\)

This means it is a one-to-one function

Step 2

To find the inverse functions we switch x and y and then solve for y

For, \(\displaystyle{x}{<}{0}\)</span>

y=x

Or, x=y

For \(\displaystyle{x}\ge{0}\)

y=2x

Or, x=2y

Or, \(\displaystyle{y}=\frac{{x}}{{2}}\)

Result:

\(\displaystyle{{f}^{{-{1}}}{\left({x}\right)}}={\left\lbrace\begin{array}{cc} {x}&{x}{<}{0}\\\frac{{x}}{{2}}&{x}\ge{0}\end{array}\right.}\)</span>

When x is less than 0 then also y is less than zero

if \(\displaystyle{x}_{{1}}\ne{x}_{{2}}{f}{\quad\text{or}\quad}{x}_{{1}},{x}_{{2}}{<}{0}\)</span>

Then, \(\displaystyle{y}_{{1}}\ne{y}_{{2}}\)

if \(\displaystyle{x}\ge{0}\) , then y is also greater than 0.

if \(\displaystyle{x}_{{3}}\ne{x}_{{4}}{f}{\quad\text{or}\quad}{x}_{{3}},{x}_{{4}}{<}{0}\)</span>

Then \(\displaystyle{2}{x}_{{3}}\ne{2}{x}_{{4}}\)

So, \(\displaystyle{y}_{{3}}\ne{y}_{{4}}\)

This means it is a one-to-one function

Step 2

To find the inverse functions we switch x and y and then solve for y

For, \(\displaystyle{x}{<}{0}\)</span>

y=x

Or, x=y

For \(\displaystyle{x}\ge{0}\)

y=2x

Or, x=2y

Or, \(\displaystyle{y}=\frac{{x}}{{2}}\)

Result:

\(\displaystyle{{f}^{{-{1}}}{\left({x}\right)}}={\left\lbrace\begin{array}{cc} {x}&{x}{<}{0}\\\frac{{x}}{{2}}&{x}\ge{0}\end{array}\right.}\)</span>