Funky functions

Method 1:

$fg(m) = f(\frac{m+1}{2m-4}) = \frac{3-\frac{m+1}{2m-4}}{3(\frac{m+1}{2m-4}) - 2} = \frac{(6m - 12)-(m+1)}{3m+3 - (4m - 8)} = \frac{5m-13}{11-m}$

$\frac{5m-13}{11-m} = \frac{1}{4}$

$4(5m-13) = 11-m$

$20m-52=11-m$

$(20+1)m = 52+11$

$21m = 63$

$m=\frac{63}{21} = \boxed{3}$

Method 2:

$fg(m) = \frac{1}{4}$

$\frac{3-g(m)}{3g(m) - 2} = \frac{1}{4}$

$12-4g(m) = 3g(m)-2$

$7g(m) = 14$

$g(m) = 2$

$\frac{m+1}{2m-4} = 2$

$m+1 = 4m - 8$

$3m = 9$

$m = \boxed{3}$