Guided Example #2 - Inverse Functions and One-to-One Functions

Start with the assumption that ${\displaystyle f\left(a\right)=f\left(b\right)}$.
${\displaystyle f\left(a\right)=\frac{a+1}{2a-3}=\frac{b+1}{2b-3}=f\left(b\right)}$by definition ${\displaystyle g\left(f\left(4\right)\right)}$.

Cross multiply
${\displaystyle (a+1)(2b-3)=(b+1)(2a-3)}$
${\displaystyle 2ab-3a+2b-3=2ab-3b+2a-3}$

Cancel like terms
${\displaystyle -3a+2b=-3b+2a}$

Cross multiply
${\displaystyle (a+1)(2b-3)=(b+1)(2a-3)}$
${\displaystyle 2ab-3a+2b-3=2ab-3b+2a-3}$

Combine like terms
${\displaystyle -5a=-5b}$

So ${\displaystyle a=b}$
Therefore, ${\displaystyle f\left(x\right)}$is a one-to-one function.

Start with the assumption that ${\displaystyle f\left(a\right)=f\left(b\right)}$.
${\displaystyle f\left(a\right)=\frac{a+1}{2a-3}=\frac{b+1}{2b-3}=f\left(b\right)}$by definition ${\displaystyle g\left(f\left(4\right)\right)}$.

Cross multiply
${\displaystyle (a+1)(2b-3)=(b+1)(2a-3)}$
${\displaystyle 2ab-3a+2b-3=2ab-3b+2a-3}$

Cancel like terms
${\displaystyle -3a+2b=-3b+2a}$

Cross multiply
${\displaystyle (a+1)(2b-3)=(b+1)(2a-3)}$
${\displaystyle 2ab-3a+2b-3=2ab-3b+2a-3}$

Combine like terms
${\displaystyle -5a=-5b}$

So ${\displaystyle a=b}$
Therefore, ${\displaystyle f\left(x\right)}$is a one-to-one function.