Finding the Inverse of a Quadratic Function
To find the inverse of the function f(x) = 3x^2 + 5, we'll need to follow these steps:
1. Replace f(x) with y: y = 3x^2 + 5
2. Switch x and y: x = 3y^2 + 5
3. Solve for y in terms of x.
Starting with the equation x = 3y^2 + 5, we'll solve for y:
x = 3y^2 + 5
x - 5 = 3y^2
(y^2) = (x - 5) / 3
y = ±sqrt((x - 5) / 3)
However, since we generally consider the principal branch (or the positive square root) for the inverse function, we will choose the positive square root:
y = sqrt((x - 5) / 3)
Therefore, the inverse function, f^(-1)(x), is:
f^(-1)(x) = sqrt((x - 5) / 3)
Do note that the domain of the original function must be restricted to non-negative values of x to ensure that the inverse is a function because the original function f(x) = 3x^2 + 5 is not one-to-one over all real numbers due to the parabola nature of y = x^2. This restriction allows the inverse to pass the horizontal line test.
