A) g(x) = - x/4 + 4 f(x) = 4x + 1/4
So
g-1 (x)
Write y for g(x)
y = -x/4 + 4
y - 4 = -x/4
4y - 16 = - x
16 - 4y = x swap x and y
16 - 4x = y = g-1(x)
f-1(x)
Write y for f(x)
y = 4x + 1/4
y - 1/4 = 4x
[ 4y - 1 ] / 4 = 4x
[ 4y - 1 ] / 16 = x swap x and y
[ 4x - 1] / 16 = y = f-1(x)
(g o f )(x) = - [4x + 1/4] / 4 + 4 = -x - 1/16 + 64/16 = 63/16 - x
(g o f)-1 (x)
Write y for (g o f)(x)
y = 63/16 - x
y - 63/16 = -x
[ 63 - 16y] / 16 = x swap x and y
[63 - 16x ] / 16 = y = (g o f )-1 (x)
( f-1 (x) o g-1(x) ) =
[ 4 [16 - 4x] - 1 ] / 16 =
[ 64 - 16x - 1 ] / 16 =
(63 / 16) - x
We can conclude that
(g o f )(x) = ( f-1 (x) o g-1(x) )