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# For each of the piecewise-defined functions, determine whether or not the function is one-to-one, and if it is, determine its inverse function. f(x)={(x , when x < 0),(2x, when x >=0):} # For each of the piecewise-defined functions, determine whether or not the function is one-to-one, and if it is, determine its inverse function. f(x)={(x , when x < 0),(2x, when x >=0):}

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Piecewise-Defined Functions asked 2021-02-24
For each of the piecewise-defined functions, determine whether or not the function is one-to-one, and if it is, determine its inverse function.
$$\displaystyle{f{{\left({x}\right)}}}={\left\lbrace\begin{array}{cc} {x}&{w}{h}{e}{n}{x}{<}{0}\\{2}{x}&{w}{h}{e}{n}{x}\ge{0}\end{array}\right.}$$</span>

## Answers (1) 2021-02-25
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>

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