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Chapter 3: Problem 71

Find the inverse of \(f(x)=\frac{2 x}{x+3} \cdot[3.1]\)

Short Answer

Expert verified
The inverse of the function \(f(x)=\frac{2x}{x+3}\) is \(f^{-1}(x)=-\frac{3x}{x-2}\).

Step by step solution

01

Swap \(x\) and \(y\)

Swap \(x\) and \(f(x)\), so the function \(f(x)=\frac{2x}{x+3}\) becomes \(x=\frac{2y}{y+3}\)
02

Cross-multiply

Cross-multiply to resolve the fraction, giving \(x(y+3)=2y\).
03

Expansion

Expand the left-hand side to obtain \(xy+3x=2y\).
04

Rearrange terms

Rearrange the equation to have only \(y\) terms on one side: \(xy-2y=-3x\).
05

Factor out \(y\)

Factor out the common term \(y\) on the left side of equation to get \(y(x-2)=-3x\).
06

Solve for \(y\)

Now we can solve the equation for \(y\), that is the inverse of the function \(f(x)\), as follows: \(y=-\frac{3x}{x-2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
Cross-multiplication is a technique used to eliminate fractions from an equation by multiplying each term in a fraction by the denominator of the opposite fraction. In the given exercise, we have an equation \[ x = \frac{2y}{y+3} \]where cross-multiplication is applied.
  • First, identify the fractions, which are \( x \) on one side and \( \frac{2y}{y+3} \) on the other side.
  • Multiply \( x \) by \( y+3 \) and set it equal to \( 2y \times 1 \).
  • This leads to the equation \( x(y+3) = 2y \) after cross-multiplying.
Cross-multiplication is particularly helpful when dealing with inverse functions because it simplifies equations from having variables in denominators, making further algebraic manipulations easier.
Factoring
Factoring is the process of breaking down an expression into simpler components or factors that, when multiplied together, give back the original expression. After cross-multiplication and expanding the equation in the given exercise, we reached the step:
\[ xy + 3x = 2y \]which needs further simplification.
  • The next step involves rearranging terms so that all terms involving the variable \( y \) are on one side. This becomes \( xy - 2y = -3x \).
  • To isolate \( y \), we factor it out from both terms on the left-hand side.
  • This results in: \( y(x - 2) = -3x \).
Factoring is a crucial strategy in solving equations as it can considerably simplify the equation, making it easier to isolate the desired variable.
Equation Solving
Equation solving is a core skill in algebra, involving steps to find the value of unknown variables. In solving equations for inverse functions, we aim to isolate the dependent variable, typically represented by \( y \) here.
  • The equation from the last factoring step was \( y(x - 2) = -3x \).
  • To solve for \( y \), divide both sides by \( x - 2 \), resulting in \( y = -\frac{3x}{x-2} \).
  • At this stage, we have found the inverse function.
This process highlights the importance of maintaining balance between both sides of the equation and systematically manipulating the equation to isolate and solve for the unknown variable.
Function Transformation
Function transformation occurs when a function is altered into its inverse or shifted in some manner across a graph. Here, we are transforming the original function \( f(x) = \frac{2x}{x+3} \) into its inverse.
  • The transformation involves swapping the roles of \( x \) and \( y \), making it possible to find the inverse.
  • The inverse function, in this case, becomes \( y = -\frac{3x}{x-2} \) or equivalently \( f^{-1}(x) = -\frac{3x}{x-2} \).
  • This transformation highlights the symmetry between a function and its inverse graphically over the line \( y = x \).
Understanding function transformation is essential for interpreting different forms and behaviors of functions, particularly in the context of problems requiring inverse functions.

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