If $A=R-\begin{Bmatrix}\frac{3}{2}\end{Bmatrix}$ and function $f: A→ A $ is defined by $f(x)=\frac{3x-2}{2x-3}$, then
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $f^{-1}(x)=f(x)$
$f(x)=\frac{3x-2}{2x-3},A=R-\left\{\frac{3}{2}\right\}$
Injectivity (One-to-one):
$⇒f(x_1)=f(x_2)$
$⇒\frac{3x_1-2}{2x_1-3}=\frac{3x_2-2}{2x_2-3}$
$⇒(3x_1-2)(3x_2-2)=(2x_1-3)(2x_2-3)$
$⇒-9x_1+4x_1=-9x_2+4x_2$
$⇒x_1=x_2$
∴ This is one-to-one function.
Surjectivity (Onto):
A function is onto if for every $y∈A$, there exists $x∈A$ such that $f(x)=y$.
$⇒y(2x-3)=3x-2$
$⇒x=\frac{3y-2}{2y-3}$
∴ f(x) is surjective.
and, $f(x)=f^{-1}(x)$