The inverse function for the function f(x) = x^2 is an important concept in mathematics, particularly in algebra and calculus. To find the inverse function, we follow a systematic approach. First, we start by replacing f(x) with y for easier manipulation:
y = x^2
Next, we want to solve for x in terms of y. By doing this, we swap x and y and rearrange the equation:
- Swap the variables:
x = y^2 - Rearrange for
y:
y = ±√x
This gives us two potential solutions for the inverse function, namely y = √x and y = -√x. This step is critical because it reveals that, technically, the original function f(x) = x^2 is not one-to-one, meaning it does not have a unique inverse across all real numbers. When x is positive or zero, we can restrict our domain to non-negative values (0 to +∞) to obtain a single-valued function for the inverse.
Hence, if we restrict the domain of the original function to [0, +∞), then the inverse function is:
f-1(x) = √x
In summary, the inverse of f(x) = x^2 is f-1(x) = √x, defined for x ≥ 0. This ensures we have a function that is both one-to-one and onto, fulfilling the necessary conditions for it to be a valid inverse.