Q. If f(x) = x^2 – 2x, g(x) = 2x + 3 and fg-1(x) = 3 then find the value of x.

Soln:

To find g^-1(x),

Let y be the image of x under g, then

y = 2x + 3

x = (y – 3)/2

Interchanging the role of x and y, we have

y = (x – 3)/2

g^-1(x) = (x -3)/2

Now,

fg^-1(x) = 3 [Given]

f[g^-1(x)] = 3

f[(x – 3)/2] = 3   [∵ g^-1(x) = (x -3)/2]

[(x -3)/2]² – 2[(x – 3)/2] = 3

[(x – 3)²/4] – (x – 3) = 3

[(x – 3)²/4] – x + 3 = 3

(x² – 6x + 9 - 4x)/4 = 3 - 3

x² -10x + 9 = 0 * 4

x² -10x + 9 = 0

x² – 9x – x + 9 = 0

x(x – 9) – 1(x – 9) = 0

(x -9)(x – 1) = 0

x = 1, 9