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Mathematics Question on Relations and functions

If f(x)=2x−33x+4f\left(x\right) = \frac{2x -3}{3x+4} then f−1(−43)=f^{-1} \left(\frac{-4}{3}\right) =

A

ZeroZero

B

34\frac{3}{4}

C

−23\frac{-2}{3}

D

None of theseNone\, of \,these

Answer

None of theseNone\, of \,these

Explanation

Solution

f(x)=2x−33x−4f\left(x\right) = \frac{2x-3}{3x-4}
Let f(x)=y=2x−33x+4 f\left(x\right) = y = \frac{2x-3}{3x+4}
On Cross multiplication, we get
⇒3xy+4y=2x−3\Rightarrow 3xy +4y = 2x-3
⇒x(3y−2)=−3−4y\Rightarrow x\left(3y -2\right) = -3 - 4y
⇒x=−3−4y3y−2\Rightarrow x= \frac{-3 - 4 y}{3y -2}
⇒x−f−1(y)=−3−4y3y−2\Rightarrow x - f^{^{-1} } \left(y\right)= \frac{-3 -4y}{3y - 2}
Put y=−43y =- \frac{4}{3} we get
f−1(−43)=−3−4×(−43)3(−43)−2f^{-1} \left(- \frac{4}{3}\right) = \frac{-3 -4 \times\left(-\frac{4}{3}\right)}{3\left(- \frac{4}{3}\right) - 2}
=−3+163−4−2=73×(−6)=−718= \frac{-3 + \frac{16}{3}}{-4 -2} = \frac{7}{ 3\times \left(-6\right)} = - \frac{7}{18}