Function composition might sound a bit complex, but it's something we see and use every day. Let's look at some simple examples to help understand this idea better.

1. Money and Finance

Think about how interest works when you save money. Imagine you start with an amount of money, which we call $P$, and you earn interest at a rate $r$.

After one year, the total amount you have is $A_1 = P(1 + r)$.

If you leave that money in for another year, your total becomes $A_2 = A_1(1 + r) = P(1 + r)(1 + r) = P(1 + r)^2$.

This shows how the interest calculation works step by step.

If we say the first step is $f(r) = P(1 + r)$ (how much you have after year one) and the second step is $g(A) = A(1 + r)$ (how much you get after year two), you can see the total after two years as $g(f(r))$.

2. Traveling

Let’s say you want to find out how far you traveled on a trip.

First, think of the distance from your house to a store as $d_1(x)$, where $x$ tells us what kind of transportation you are using (like driving or biking).

Then, consider the distance from the store to your friend's house as $d_2(y)$, based on the route you take.

So, if you drive to the store and then bike to your friend's place, you can find the total distance traveled as $d_2(d_1(x))$.

3. Computers and Programming

Function composition is a big part of programming too!

Imagine you have a function that takes in user information, which we can call $f(input)$. Then, you have another function that saves that processed information, called $g(output)$.

You can show how this works together with $g(f(input))$. This shows how information goes from one step to the next.

4. Health and Fitness

When you keep track of calories, you might have one function $f(x)$ that shows how many calories you burn based on the type of workout $x$.

Then, you have another function $g(y)$ that tells you how many calories you eat based on your diet $y$.

By using $g(f(x))$, you can figure out if you are gaining or losing weight based on your calories burned and consumed.

Conclusion

So, function composition isn't just a fancy math idea—it's something we use in everyday life! Whether we're talking about money, travel, technology, or health, understanding how functions connect can help us make better choices.