# 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):}

Question
Piecewise-Defined Functions
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>

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>

### Relevant Questions

For each of the piecewise-defined functions in determine whether or not the function is one-to-one, and if it is, determine its inverse function.
$$\displaystyle{f{{\left({x}\right)}}}={\left\lbrace\begin{array}{cc} {x}^{{2}}&{w}{h}{e}{n}{x}{<}{0}\\{x}&{w}{h}{e}{n}{x}\ge{0}\end{array}\right.}$$
Determine whether the following function is a polynomial function. If the function is a polynomial​ function, state its degree. If it is​ not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term.
$$g(x)=3-\frac{x^{2}}{4}$$

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$$a^m inH$$
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$$a^m inH$$
f(-3),f(0),f(2),f(3),f(5)

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Determine whether $$g(x)=\frac{x^{3}}{2} -x^{2}+2$$ is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term.
$$\displaystyle{f{{\left({x}\right)}}}={\left\lbrace{4},-{4}{<}{x}{<}-{2},{2}{x}-{4},-{1}{<}{x}{<}{2},{3}{x},{2}\le{x}{<}{5}\right\rbrace}$$
a. The domain ____ is used when graphing the function $$f(x)=2x-4$$.
b. The equation ____ is used to find $$f(4)=12.$$
$$G(x)=2(x-3)^{2}(x^{2}+5)$$