Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
0 votes

Let \(f : R - \{ - \frac {4} {3} \} \to R \) be a function defined as \(f(x)= \frac {4x} {3x+4} \) .The inverse of \(f\) is the map \(g\): Range \(f \to R - \{ - \frac {4} {3} \} \) given by

 \begin{array}{1 1}(a)\;\;\;\;\; g(y)= \frac {3y} {3-4y} & (b)\;\;\;\;\; g(y)= \frac {4y} {4-3y}\\(c)\;\;\;\;\; g(y)= \frac {4y} {3-4y} & (d)\;\;\;\;\; g(y)= \frac {3y} {4-3y}\end{array}

Can you answer this question?

1 Answer

0 votes
  • A function $g$ is called inverse of $f:x \to y$, then exists $g:y \to x$ such that $ gof=I_x\;and\; fog=I_y$, where $I_x, I_y$ are identify functions.
  • Given two functions $f:A \to B $ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $ gof (x)=g(f(x))\;for\; all \;x \in A$
Given a function $f:R-${$\frac{-4}{3}$}$\to R$ defined by $f(x)=\large \frac{4x}{3x+4}$.
$\textbf {Step 1: To calculate the inverse of } f, \textbf {we must first define} \;g(y):$
Let $y = f(x) =\large \frac{4x}{3x+4}$.
$\Rightarrow 3xy + 4y = 4x \rightarrow x =\large \frac{4y}{4-3y}$
Let us now define a function $g:$Range $f \to R-\{\frac{-4}{3}\}$ given by $g(y)=\large \frac{4y}{4-3y}$
$\textbf {Step 2: Calculating} \;gof$:
We know that $(gof) (x)=g(f(x))$
$\Rightarrow gof =g(\frac{4x}{3x+4})$
$\Rightarrow gof =\Large\frac{4(\frac{4x}{3x+4})}{4-3(\frac{4x}{3x+4})}$
$\Rightarrow gof =\Large \frac{16x}{12x+16-12x}=\frac{16x}{16}$ = $x$
$\textbf {Step 3: Calculating} \;fog$:
We know that $(fog) (y)=f(g(y))$
$\Rightarrow fog$ $=f(\frac{4y}{4-3y})$
$\Rightarrow fog$ $=\Large\frac{4(\frac{4y}{4-3y})}{3(\frac{4y}{4-3y})+4}$
$\Rightarrow fog$ $=\Large \frac{16y}{12y+16-12y}=\frac{16y}{16}$ = $y$
$\textbf {Step 4: Calculating} \;\;f^{-}\; \textbf {from}\; \;gof = fog$
$gof=I_{R-\{\frac{4}{3}\}}$ and $fog=I_{Range f}$
Therefore $ f^{-1}=g$ defined by $g:Range f \to R-\{\frac{4}{3}\}$ where $g(y)=\large \frac{4y}{4-3y}$
Therefore $(B) \; g(y)=\large \frac{4y}{4-3y}$ is the correct answer.
answered Feb 26, 2013 by meena.p
edited Jul 17, 2014 by balaji.thirumalai

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App