Prove that
Question:

If $\mathrm{y}=\mathrm{f}(\mathrm{x})=\frac{3 \mathrm{x}+1}{5 \mathrm{x}-3}$, prove that $\mathrm{x}=\mathrm{f}(\mathrm{y})$

Solution:

Given: $y=f(x)=\frac{3 x+1}{5 x-3}$

Need to prove: $x=f(y)$

Replacing x by y in the function,

$f(y)=\frac{3 y+1}{5 y-3}$

Now, given in the problem that $y=f(x)$

$f(y)=\frac{3 f(x)+1}{5 f(x)-3}$

$\Rightarrow f(y)=\frac{3 \frac{3 x+1}{5 x-3}+1}{5 \frac{3 x+1}{5 x-3}-3}$

$\Rightarrow f(y)=\frac{9 x+3+5 x-3}{15 x+5-15 x+9}$

$\Rightarrow f(y)=\frac{14 x}{14}=x$

$\Rightarrow x=f(y)$ [Proved]