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):}

Question
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>
0

Relevant Questions

asked 2020-11-08
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.
\(\displaystyle{f{{\left({x}\right)}}}={\left\lbrace\begin{array}{cc} {x}^{{2}}&{w}{h}{e}{n}{x}{<}{0}\\{x}&{w}{h}{e}{n}{x}\ge{0}\end{array}\right.}\)
asked 2021-06-03
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.
\(g(x)=3-\frac{x^{2}}{4}\)
asked 2021-01-13

Evaluate the piecewise defined function at the indicated values.
\(a^m inH\)
\(f(x)= \begin{array}{11}{5}&\text{if}\ x \leq2 \ 2x-3& \text{if}\ x>2\end{array}\)
\(a^m inH\)
f(-3),f(0),f(2),f(3),f(5)

asked 2021-06-03
Find the discriminant of each equation and determine whether the equation has (1) two nonreal complex solutions, (2) one real solution with a multiplicity of 2, or (3) two real solutions. Do not solve the equations. \(7x^{2} - 2x - 14 = 0\)
asked 2021-06-04
Determine whether \(F(x)=5x^{4}-\pi x^{3}+\frac{1}{2}\) 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.
asked 2021-02-26
The top string of a guitar has a fundamental frequency of 33O Hz when it is allowed to vibrate as a whole, along all its 64.0-cm length from the neck to the bridge. A fret is provided for limiting vibration to just the lower two thirds of the string, If the string is pressed down at this fret and plucked, what is the new fundamental frequency? The guitarist can play a "natural harmonic" by gently touching the string at the location of this fret and plucking the string at about one sixth of the way along its length from the bridge. What frequency will be heard then?
asked 2021-05-20
Assume that a ball of charged particles has a uniformly distributednegative charge density except for a narrow radial tunnel throughits center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton any where along the tunnel or outside the ball. Let \(\displaystyle{F}_{{R}}\) be the magnitude of the electrostatic force on the proton when it islocated at the ball's surface, at radius R. As a multiple ofR, how far from the surface is there a point where the forcemagnitude is 0.44FR if we move the proton(a) away from the ball and (b) into the tunnel?
asked 2021-05-01
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.
asked 2021-02-21

Use the piecewise-defined function to fill in the bla
.
\(\displaystyle{f{{\left({x}\right)}}}={\left\lbrace{4},-{4}{<}{x}{<}-{2},{2}{x}-{4},-{1}{<}{x}{<}{2},{3}{x},{2}\le{x}{<}{5}\right\rbrace}\)
a. The domain ____ is used when graphing the function \(f(x)=2x-4\).
b. The equation ____ is used to find \(f(4)=12.\)

asked 2021-05-25
Determine whether the following function is a polynomial function. If the function is a polynomial​ function, state its degree. If not why? Write the polynomial in standard form. Then identify the leading term and the constant term.
\(G(x)=2(x-3)^{2}(x^{2}+5)\)
...