To algebraically verify that two functions, let's call them f(x) and g(x), are inverses of each other, you would need to verify two conditions.
1. Check if applying g(x) within the function f, represented as f(g(x)), gives us the same as the initial input, that is x. If this condition holds, it indicates that g(x) is the inverse of f(x) because applying g on any value and then feeding the result into the function f brings you back to this initial value.
2. Similarly, check if applying f(x) within the function g, represented as g(f(x)), gives us the same as the initial input, x. This means that applying f on any value and then applying function g to the result brings you back to the original value. If this is true, it shows that f is the inverse of g.
If both these checks are successful and you get back your original input on both accounts, you can conclusively say that f(x) and g(x) are inverses of each other. Algebraically, this is proven by the fact that applying a function and its inverse gives you back the original argument that you began with.