Question

Find whether the function $f\left(x\right)=\frac{x^2+4x+30}{x^2-8x+18}$, is one-one.

Answer 1

Let us approach trial and error method.

Let $f\left(x\right)=2$

or, $\frac{x^2+4x+30}{x^2-8x+18}=2$

or, $2x^2-16x+36=x^2+4x+30$

or, $x^2-20x+6=0$

or, $x=\frac{20\pm \sqrt{{20}^2-24}}{2}=10\pm \sqrt{94}$.

So, we have two distinct real values of $x$ for which $f\left(x\right)$ has one value.

So, for $x_1\ne x_2\nRightarrow f\left(x_1\right)\ne f\left(x_2\right)$.

So $f\left(x\right)$ is not one-one.