Q. h(x) = x/(2-x) and g(x) = ax-5 are two functions. IF goh(4) = -13, find the values of 'a' and g-1(1).

Soln:

h(x) = x/(2-x)

So, h(4) = 4/(2-4) = -2

We have, goh(4) = -13

or, g[h(4)] = -13

or, g(-2) = -13       [Since, h(4) = -2]

a*(-2) – 5 = -13,   [Since, g(x) = ax-5]

-2a-5 = -13

2a = 13 – 5

a = 8/2

\ a = 4.

To find g^-1(x).

Let y be the image of x under g. Then,

y = ax – 5

or, x = (y + 5)/a

Interchanging the role of x and y, we get,

y = (x+5)/a

\ g^-1(x) = (x+5)/a = (x+5)/4 [Since, a = 4]

So, g^-1(1) = (1+5)/4 = 3/2

Hence, a = 4 and g^-1(1) = 3/2 Ans.