If f(x) =1/(x² - 8)²,find the expression for inverse function g(f) =f^(-1) (x)

Answer 1

To find the inverse of a function, we usually replace f(x) with y and then swap the x and y variables. So let's replace f(x) with y:

y = f(x) = 1/(x² - 8)²

Now, swap x and y:

x = 1/(y² - 8)²

To find the inverse function, let's solve for y:

(x² - 8)² = 1/y
√((x² - 8)²) = √(1/y)
|x² - 8| = 1/√y
x² - 8 = ±(1/√y)

We consider both the positive and negative square root:

x² - 8 = 1/√y or x² - 8 = -1/√y

Solving for y:

1/√y = x² - 8 or -1/√y = x² - 8

√y = 1/(x² - 8) or √y = -1/(x² - 8)

Squaring both sides:

y = (1/(x² - 8))² or y = (-1/(x² - 8))²

Simplifying:

y = 1/(x² - 8)² or y = 1/(x² - 8)²

Therefore, the expression for the inverse function g(f) = f^(-1)(x) is:

g(f) = (1/(x² - 8)²)