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Please Solve RD Sharma Class 12 Chapter Function Exercise 2.4 Question 16 Maths Textbook Solution.

Answers (1)

Answer:

                f^{-1}\left ( x \right )=\frac{4}{4-3x}

Given:

                f:R-\left \{ \frac{-4}{3} \right \}\rightarrow R-\left \{ \frac{4}{3} \right \}\:is given by \: f\left ( x \right )=\frac{4x}{3x+4}

Injectivity,

Let   x,y\: \epsilon \: R-\left \{ \frac{-4}{3} \right \} be such that f\left ( x \right )=f\left ( y \right )

                \frac{4x}{3x+4}=\frac{4y}{3y+4}

                4x\left ( 3y+4 \right )=4y\left ( 3x+4 \right )

                12xy+16x= 12xy+16y

                16x=16y

                x=y

fis a one-one function.

Let y\: be an arbitrary element of R-\left \{ \frac{4}{3} \right \}.

Then      f\left ( x \right )= y

                \frac{4x}{3x+4}=y

                4x=3xy+4y

                4x-3xy=4y

                x=\frac{4y}{4-3y}

As           y\: \epsilon \: R-\left \{ \frac{4}{3} \right \},\frac{4y}{4-3y}=\frac{-4}{3}

Also,      \frac{4y}{4-3y}\neq \frac{-4}{3} because,  \frac{4y}{4-3y}= \frac{-4}{3}

                12y=-16+12y\Rightarrow 0=-16

Which is not possible.

Thus,

                x=\frac{4y}{4-3y}\: \epsilon \: R-\left \{ \frac{-4}{3} \right \}

                f\left ( x \right )=f\left ( \frac{4x}{3x+4} \right )=\frac{4\left ( \frac{4y}{4-3y} \right )}{3\left ( \frac{4y}{4-3y} \right )+4}

                                =\frac{16y}{12y+16-12y}=\frac{16y}{16}

                                y

So every element in R-\left \{ \frac{4}{3} \right \} has pre image in R-\left \{ \frac{-4}{3} \right \}

Hence f is onto.

Now, x=\frac{4y}{4-3y}, replacing x by f^{-1} and y by x,

                f^{-1}\left ( x \right )=\frac{4}{4-3x}

Posted by

infoexpert27

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