Why the Square Root of x is Not the Inverse of x^2

Understanding the relationship between the functions g(x) sqrt{x} and f(x) x^2 requires a deep dive into the domains, ranges, and the nature of these functions. Contrary to what might initially be inferred, g(x) sqrt{x} is not the inverse of f(x) x^2 due to key mathematical properties that must be considered.

Domain and Range

The function f(x) x^2 is defined for all real numbers, but it results in a non-negative output. This means that the range of f(x) x^2 is [0, infty). The function g(x) sqrt{x}, on the other hand, is only defined for non-negative real numbers, which means its domain is [0, infty). Both functions map non-negative numbers back to non-negative numbers, but the domains and ranges do not match up in a one-to-one correspondence necessary for inversion.

Behavior of the Functions

The function f(x) x^2 is not one-to-one. This is because it maps multiple inputs to the same output. For instance, both 2 and -2 yield the same output 4. That is, f(2) 4 and f(-2) 4. For a function to have an inverse, it must be a bijective function, which means it must be both injective (one-to-one) and surjective (onto). Since f(x) x^2 is not one-to-one, it does not have an inverse function defined over all real numbers.

Regarding the inverse relation, if we consider g(f(x)) sqrt{x^2}, we find that it equals |x|, not x itself. This means that g(x) sqrt{x} does not return the original input x for all x; it returns the non-negative counterpart of x.

Conclusion

For g(x) sqrt{x} to be the inverse of f(x) x^2, the domain of x must be restricted to non-negative values. In this case, g(f(x)) x holds true if x is non-negative. Therefore, the functions are inverses of each other if we restrict the domain of f(x) to non-negative numbers, resulting in:

In summary, sqrt{x} is the inverse of x^2 only when x is restricted to non-negative values. This conclusion emphasizes the importance of understanding the domain and the nature of functions in establishing inverse relationships.

Solution

To address the issue of g(x) sqrt{x} not being the inverse of f(x) x^2, we must consider the domain of x. The domain of f(x) x^2 is all real numbers. If we restrict the domain of f(x) to non-negative numbers, then we have a new function, and its inverse is g(x) sqrt{x}.