To confirm that two functions, f and g, are inverses of each other, we need to show that applying one function to the result of the other returns us to the original input. This requires evaluating two specific expressions:
- f(g(x)) = x
- g(f(x)) = x
Let’s break down the process step by step:
Step 1: Evaluate f(g(x))
To find f(g(x)), start by substituting g(x) into the function f. For example, if f(x) = 2x + 3 and g(x) = (x – 3) / 2, then:
f(g(x)) = f((x – 3) / 2)
This becomes:
f(g(x)) = 2((x – 3) / 2) + 3
Simplifying this gives:
f(g(x)) = x – 3 + 3 = x
Step 2: Evaluate g(f(x))
Next, evaluate g(f(x)) by substituting f(x) into the function g. Using the same example:
g(f(x)) = g(2x + 3)
This becomes:
g(f(x)) = ((2x + 3) – 3) / 2
Simplifying this gives:
g(f(x)) = (2x) / 2 = x
Conclusion
Since we have shown both:
- f(g(x)) = x
- g(f(x)) = x
This confirms that the functions f and g are indeed inverses of each other. Being able to rely on the relationship between inverse functions is crucial in mathematics, especially in calculus and algebra, as it helps in solving equations and understanding function behavior.