find the inverse of f(x)=x^2-8x, x is greater than or equal to 4
To find the inverse of a function, we need to switch the roles of x and y in the original function and then solve for y.
1. Start with the original function: f(x) = x^2 - 8x.
2. Switch x and y: x = y^2 - 8y.
3. Rearrange the equation to solve for y: y^2 - 8y - x = 0.
4. This quadratic equation can be solved using the quadratic formula. The quadratic formula is given by:
y = (-b ± sqrt(b^2 - 4ac))/(2a).
In our case, a = 1, b = -8, and c = -x.
Plugging these values into the quadratic formula, we get:
y = (8 ± sqrt((-8)^2 - 4(1)(-x))) / (2(1)).
Simplifying further:
y = (8 ± sqrt(64 + 4x)) / 2.
y = (8 ± sqrt(4(x + 16))) / 2.
y = (8 ± 2√(x + 16)) / 2.
5. Simplify the expression:
y = 4 ± √(x + 16).
Therefore, the inverse of f(x) = x^2 - 8x (where x is greater than or equal to 4) is given by:
f^(-1)(x) = 4 ± √(x + 16).