A-2 Rational Function

In f( $\frac{x - 3}{5 - x}$ ), we want to make this function into f(1)

Thus, solving the problem we have to equate $\frac{x - 3}{5 - x}$ = 1, and we get x = 4

Then substituting thus value of x in the expression $\frac{3x - 2}{x+1}$ , we have

$\frac{3x - 2}{x - 2}$ = $\frac{3(4) - 2}{(4) - 2}$ = 5

Hence the answer is $\boxed{5}$

I prefer an alternative means. We rewrote the original equation to $f(-1+\frac{2}{5-x}) = 3+\frac{4}{x-2}$ Let $y = -1+\frac{2}{5-x}$ . Rewrote it to $x = 5-\frac{y+1}{2}$ , then plug in $y$ to the LHS and $x$ to the RHS. This will give $f(y) = 3+\frac{8}{5-y}\Rightarrow f(x)=3+\frac{8}{5-x} = \frac{23-3x}{5-x}$ Here, just plug in $x = 1$ and the answer will be $f(1) = 3+\frac{8}{5-1} = 3+2 = 5$