If f(x) =1/(x² - 8)²,find the expression for inverse function g(f) =f^(-1) (x)
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)
To find the expression for the inverse function g(f) = f^(-1)(x), we need to solve the equation for x in terms of f.
Let's start by replacing f(x) with y for simplicity:
y = 1 / (x² - 8)²
To find the inverse, we swap the roles of x and y and solve for y:
x = 1 / (y² - 8)²
Now, let's solve for y in this equation. To do this, we'll multiply both sides of the equation by (y² - 8)²:
x(y² - 8)² = 1
Next, we can take the square root of both sides:
√(x(y² - 8)²) = √1
y² - 8 = ±√x
Simplifying further:
y² = 8 ± √x
Taking the square root of both sides:
y = ±√(8 ± √x)
Therefore, the expression for the inverse function g(f) = f^(-1)(x) is:
g(f) = ±√(8 ± √x)