Hey, everyone, and welcome back. So over the course of the past couple of videos, we've been talking about the composition of functions. Now in this video, we're going to be taking a look at function decomposition. Decomposition is basically just the reverse of function composition, and this process can often be a bit tricky because it's hard to know when reversing this process what the functions are going to look like. We're used to starting with two functions and composing them into one thing, but if we start with the composed function and go backwards, that step can be a bit tricky. But don't worry about it because in this video, we're going to look at some strategies you can use to do this, and you might find actually after doing this a few times that this process can actually be kind of fun because you can be creative with your answers here. So let's get right into this.
Now there are many correct answers when it comes to decomposing a function. And to understand this best, I think it's overall best to just look at an example. In this example we have below, we're told to express the function \( h(x) \) as h ( x ) = x - 2 as a composition of two functions \( f \) and \( g \), so that \( h(x) = f(g(x)) \). This is saying that the function \( h(x) \) could also be written as \( f(g(x)) \), and we're asked to find what the individual functions would look like before this composition.
One common strategy for doing this is to look at your inside function and set that equal to whatever is inside an operation. For example, if you had a fraction, you could set \( g(x) \) equal to the denominator of that fraction. If you had a square root, you could set \( g(x) \) equal to what's inside the square root, which would be \( x - 2 \). So in this case, we could say that g ( x ) = x - 2 and then \( f(x) \) would be the square root, which is on the outside but needs to be applied to some input \( x \), so we define \( f(x) \) as x . This method is one of the ways that we could decompose this function into \( f(x) \) and \( g(x) \), and it's one of the most common strategies.
Another strategy we could have taken is to set \( g(x) \) equal to the entire expression \( \sqrt{x - 2} \) and \( f(x) \) would just be \( x \). This may seem overly simple, but this is technically a correct answer also because if we recompose these functions, we get what we started with. You could even try more unusual decompositions once you get proficient with the concept. For example, set g ( x ) = x - 2 - 1000 and \( f(x) \) could then be \( x + 1000 \). Reversing this gives you back \( \sqrt{x - 2} \), showing creative but valid decompositions.
All of these solutions that we see here are perfectly acceptable ways to decompose this function. Some of these are a little bit ridiculous. The most common case that you'll see is the one that we have over here, but either way, this is how you can do basic function decomposition. So hopefully, you found this video helpful. Thanks for watching.