To confirm that the functions f and g are inverses of each other, we need to show that
- f(g(x)) = x
- g(f(x)) = x
Let’s take each of these statements one at a time:
Step 1: Show that f(g(x)) = x
To start, we will substitute g(x) into the function f. Let’s assume:
- g(x) = some expression
- f(g(x)) = substitute the expression of g into f
If after simplification of f(g(x)), we arrive at x, this confirms the first part.
Step 2: Show that g(f(x)) = x
Next, we perform the reverse operation by substituting f(x) into the function g. Again, we assume:
- f(x) = some expression
- g(f(x)) = substitute the expression of f into g
After a similar process of simplification, if we find that g(f(x)) = x, then we have confirmed the second part.
In conclusion, if both conditions hold true, we have successfully shown that f and g are indeed inverse functions of one another.