A function is called bijective if it is one-to-one and onto.
To check one to one, let x1 = x2 , where x1, x2 belongs to R.
so, we can say that:
[tex]3x_1+4=3x_2+4[/tex]and
[tex]x_1-2=x_2-2[/tex]Given the condition that x1 and x2 are not equal to 2. So, the above equation implies that
[tex]\frac{3x_1+4}{x_1-2}=\frac{3x_2+4}{x_2-2}\Rightarrow f(x_1)=f(x_2)[/tex]So, it is a one-to-one function.
For onto function, let y= (3x + 4) / (x-2). Now, interchange x and y, and solve the equation fo y:
[tex]\begin{gathered} x=\frac{3y+4}{y-2} \\ \Rightarrow xy-2x=3y+4 \\ \Rightarrow xy-3y=2x+4 \\ \Rightarrow y(x-3)=2x+4 \\ \Rightarrow y=\frac{2x+4}{x-3} \end{gathered}[/tex]Here, the domain of the inverse of the function is R - {3}. But the function is R - {2} => R. So, it is not onto.
thus, the given function is not a bijection.
