Function Composition - Video Tutorials & Practice Problems

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Function Composition

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Hey everyone and welcome back. So we've been talking about function operations in recent videos and in this video, we're going to be looking at function compositions. Now, when it comes to composition of functions, this process can oftentimes be a bit difficult when you first encounter it. But in this video, we're going to take a look at some examples of things that we've seen already up to this point and see how we can build off of this to really understand this concept of composing a function. So let's get right into this. When it comes to a function composition, it's like evaluating a function at a certain number. But instead of replacing the inside of the function with a variable, you replace the function with another function. Now, before into composition, let's remind ourselves how we can evaluate functions at certain values. So in this case, we have the function X squared plus three X minus 10 and we're asked to evaluate this at F of seven. So in this case, we're going to take all of our X's and replace them with seven. So X squared will become seven squared and then we'll have three times seven minus 10. Now, seven squared is 49. And then this is gonna be plus three times seven, which is 21 minus 10. And if you go ahead and combine all these numbers here, you should get 60. So when you evaluate a function at a certain number, you're going to end up getting a number as your final result. Now when it comes to composing a function, it's very similar to this process of evaluating a function except rather than replacing the X with a number, we're going to replace X with another function. So in this case, we have F of X which is X squared plus three, X minus 10. And then we have G of X which is X minus two. And if we want to find F of G of X, notice we have G of X on the inside. So that means we take all of these X's and we replace them with G of X. So in this case, we're going to have the F of G of X is equal to X minus two squared plus three times X minus two minus 10. Now I won't bore you with the details of trying to simplify everything and foil everything out. But if you were to simplify this entire expression, you should end up with X squared minus X minus 12. So this is what we get when we simplify F of G of X and notice that the result is a function whenever we're doing a function composition. So this is the main idea when composing a function. Now something to note here is that whenever you see the situation where you have F of G of X, this composition, you can also write this as F of G of X. This is another way to write this notation. So it kind of looks like the word fog where we have this open circle in between F and G and then the X here. So this is the same thing as putting G inside of F. And when we, whenever you see this notation, the letter, the first letter that you see this F here, this is going to be the outside function. So like the function that you have on the outside, whereas the second letter that you see this G here is going to be the inside function. So that's just something to keep in mind. Now, let's see at this point if we can solve an example. So in this example, we have the function F of X is equal to X plus four and G of X is equal to X squared minus three. And we're asked to find the following composite functions and fully simplify our answer. So in this first situation, we have F of G of X. So what this means is we need to put G of X inside of F of X. So if we do this, we're going to take whatever X's we see and replace it with the G of X. So X is going to be replaced with what we have here, which is X squared minus three. And then we're going to have this plus four afterwards. So this right here is going to be our composite function. And if I go ahead and simplify this, well, this minus three and this positive four will reduce or combine. So negative three plus four is one. So we should end up with X squared plus one. And this right here is the composite function. And the answer for F of G of X. Now let's see how we can solve this example. For part B for part B, we're asked to find G of F of X and notice in this function composition F of X is the inside function. So that means we're going to do things backwards and we're gonna take our F of X here and we're going to plug it inside of G of X. So in this case, we're going to replace the X with this expression here. So X squared minus three will become X plus four squared minus three. Now, we need to do a bit more work here because notice we have X plus four squared. So X plus four squared is the same thing as X plus four times X plus four minus three. Now, at this point, I can use the foil method. So we have X times X, which will give us X squared, we have X times four, which will give us four X, we have four times X which is four X and then we have four times four, which is 16. And then this whole thing will be minus three. Now, at this point, we can combine these four X's here to give us eight X. We have X word plus eight X. Let me write that eight a little bit better. And then we have plus 16 minus three and 16 minus three is 13. So this right here is going to be our polynomial which we get when we simplify G of F of X. So dealing with the compositions in this problem G of F of X would look like this and F of G of X would look like that. So this is how you can do composition of functions. Hopefully, you found this video helpful and let me know if you have any questions.

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Problem

Problem

Given the functions $f\left(x\right)=\sqrt{x+4}$ and $g\left(x\right)=\left(x-2\right)^2-4$ find $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$

Hey everyone and welcome back. So in the last video, we talked about compositions of functions and in this video, we're going to be taking a look at how we can evaluate functions which have been composed. Now, the interesting thing about evaluating composed functions is that there are actually multiple different methods you can use to do this. One of these methods is actually kind of a shortcut method but it doesn't work every single time. So without further ado let's get right into this. So when dealing with composed functions, you'll often be asked to evaluate the function at a certain number. And to do this, one of the methods you can use actually builds off the last video where we can those two functions together. So the first method is taking two different functions and composing them like we saw in the last video and then taking whatever function we get from this composition and plugging in the number into the final function. So this is one way we can evaluate composed functions. And to understand this, let's take a look at an example. So in this example, we have the function F of X is equal to X squared and G of X is equal to X minus one, we're asked to find F of G of X and then evaluate F of G of three. So to solve this, what I'm first going to do is find F of G of X. And this would require me to take our inside function G of X and plug it into F of X. So in this case, since we see that G of X is X minus one and F of X is X squared, we're going to end up with X minus one squared as our composed function. Now, I do need to simplify this a bit because we typically want to get our most simplified answer. So X minus one squared is the same thing as X minus one times X minus one. Now I can do the foil method where I multiply these X's giving us X squared, we have X times negative one which is negative X, we have negative one times X which is also negative X and the negative one times negative one, which is positive one. Now, from here, I can combine the two negative XS giving us X squared minus two X plus one. So this right here is what we get when we do the composition F of G of X. Now we're asked to take this function here and evaluate it at three because we're trying to figure out what F of G of three is So to find F of G of three, notice how I can just take this X that we have here and replace it with three. Meaning all these X's can be replaced with three. So we're going to end up having three squared minus two times three plus one and three squared is the same thing as nine and we have minus two times three and two times three is six plus one, nine minus six is three. So we end up with three plus one, which is four. So this right here is the solution for F of G of three. And this is one of the methods we can use for evaluating composed functions. Now, you may recall at the start of this video, I mentioned that there's a potential shortcut method we could use. So let's take a look at what this method would look like if we were. So the same problem. So for this method, what you want to do is first figure out what your inside function is and then you want to evaluate that inside function at the number you're looking at once you found what number this is equal to, you take that number and plug it into your final function to evaluate the composed function. Now, that might sound a bit confusing. But let's take a look at this example here and see if we can do this step by step. So we have the same two functions that we had before where we're trying to figure out what F of G of three is now, rather than finding F of G of X first. Instead, I'm going to figure out what G of three is first. So G of three is going to give us three minus one because we just replaced this X with a three and three minus one is equal to two. So now that we found G of three, we can evaluate F of G of three because notice that I did this first step where we found G of our number. So now I just need to evaluate F of G of that number and F of G of three. Well, since G of three is two, this is the same thing as F of two. And for F of two, all I need to do is take two and plug it into our function F of X. So F of X is X squared, meaning that F of two would be two squared and two squared is simply equal to four. So notice when using the second method, we got the same answer that we did for the first method except we did this much faster. So the question becomes at this point, why don't we just use this method every single time? Well, the reason for this is because oftentimes when solving these problems, you'll be asked to find F of G of X first before evaluating. And whenever this happens, you have to find the original composition before you can plug in your number. So this is a shortcut method that only works for some problems but be very wary of whether or not you're asked to find F of G FX first. So this is how you can evaluate composed functions. Hopefully, you found this video helpful. Let me know if you have any questions.

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Problem

Problem

Given the functions $f(x)=x+3$ and $g(x)= x^2$ find $(f∘g)(2)$and $(g∘f)(2)$.

A

$(f∘g)(2)=5$ ; $(g∘f)(2)=25$

B

$(f\circ g)(2)=7;(g\circ f)(2)=4$

C

$(f∘g)(2)=7$ ; $(g∘f)(2)=25$

D

$(f∘g)(2)=1$ ; $(g∘f)(2)=1$

6

concept

Domain Restrictions of Composed Functions

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Hey everyone and welcome back. So in the previous video, we talked about how we could evaluate composed functions. Now, in this video, we're going to be taking a look at finding the domain of functions which have been composed. Now, this process can oftentimes be a bit tricky because it can be difficult to recognize how the domain of the functions will behave when they're separate versus how the domain acts when the functions are composed together. But don't worry about it because in this video, we're going to be going over an example that I think will really clear this entire concept up without further ado let's get right into this. Now, when finding the domain of composed functions, there are a few steps you can follow and I think it's best to see these steps by looking at an example. So given below is an example where we have two functions, we have F of X is equal to one over X minus two and G of X is equal to the square root of X. We're asked to determine the composition F of G of X and find its domain. Now keep in mind this F of G of X that we see here is the same thing as F of G of X where G of X is the inside function. Now your first step when dealing with these types of problems is to find the X values not defined for G of X. So what you first want to do is find any restrictions on the inside function. Now, if I look at our G of X function, we can see here that G of X, the inside function is equal to the square root of X. Now recall that we've talked about in previous videos how the square root of X does have restrictions because we cannot have a negative number underneath the square root, you cannot take the square root of something that's negative. So in this case, we would say that X has to be greater than or equal to zero because X cannot be negative. So this would be the restrictions for our function G of X. Now, once you've found these restrictions, your next step should be to find any X values that make G of X not defined for F of X. And if you're not sure what this means, basically what this is saying is if we take G of X and we put it inside of F of X, what restrictions are we going to have on that entire function when we compose the two together? So to figure this out, what I'm going to do is actually compose these functions. So we'll have F of G of X. Now, in doing this step, what I can do is take our function G of X and I can place it inside of the function F of X. So I'll just replace this X with the function we have here. So rather than having one over X minus two, our function F of G FX is going to be one over the square root of X minus two. Because the square root of X is this function G of X. Now we already determined the restrictions for the square root of X which is that X has to be positive. But notice that when we plug G into F, we get this new fraction that we see here, this new kind of equation. And when we do this, we see that for a fraction, the denominator cannot equal zero. That's another restriction that we have. So we can say that this whole denominator square root of X minus two cannot be equal to zero. Now, what it can do from here is solve for X in this mini equation by adding two on both sides that will get the positive and negative two to cancel, giving me that the square root of X cannot equal two. And from here, I can take both sides of this equation and square it. This will allow me to cancel the square in the square root giving me the X cannot equal two squared which is four. So this right here is the restriction for our function F of G of X. And notice at this point, we have found the restriction for G of X which was the first step. And we found the restriction for F of G of X, which was the second steps. Since we have found these two restrictions, we can now write out what our domain is. So the domain for our composed function F of G of X is that we can go from zero and we are including the zero because X can be greater than or equal to zero. So we can go from zero all the way up, up to four, not including four. And then we can go from four to infinity. So basically, we can have any number as long as it's above zero and not equal to four. So this right here would be the domain of our function F of G of X. And notice how we found first the domain for G of X, then we found F of G of X domain and then we combine these together to get the total domain. So this is how you can find the domain of composed functions. Hopefully, you found this video helpful and thanks for watching.

7

Problem

Problem

Given the functions $f(x) = x^2$ and $g(x)=\sqrt{x-8}$ find $(f∘g)(x)$and determine its domain.

A

$(f∘g)(x)=x-8$ ; $Dom:(-\infty,\infty)$

B

$(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:(-∞,∞)$

C

$(f∘g)(x)=x-8$ ; $Dom:[8,∞)$

D

$(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:[8,∞)$

8

concept

Decomposition of Functions

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Hey, everyone and welcome back. So over the course of the past couple of videos, we've been talking about composition of functions. Now, in this video, we're going to be taking a look at function decomposition. Now, decomposition is basically just the reverse of function composition. And this process can oftentimes be a bit tricky because it's hard to know when reversing this process what the functions are going to look like because 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 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 it's overall best to just look at an example. So in this example, we have below, we're told to express the function H of X is equal to the square root of X minus two as a composition of two functions F and G so that H of X is equal to F of G of X. So basically what this is saying is that the function that we have here H of X could also be written as F of G of X. And we're asked to find what the individual functions would look like before this composition. So what we need to do is figure out what our functions F of X and G of X are going to look like. And one of the common strategies for doing this is to look at your inside function and set that equal to whatever is inside an operation. So for example, if you had a fraction, you could set this G of X equal to the denominator of that fraction. If you have a square root, you could set this, this G of X equal to what's inside the square root, which would be this X minus two. So this is a strategy you could take. So in this case, we could say that G of X is equal to the square or excuse me, it would just be equal to X minus two because it's everything inside the square root. And then F of X would be the square root which is on the outside. But we can't just write a square root by itself because that doesn't mean anything. So we have to make this the square root of X. So this right here is one of the ways that we could decompose this function into F of X and G of X. And this is going to be one of the most common strategies that you see is to do it like this. And we know that this is correct because notice if you were to take this function G of X and plug it in for F of X, the square root of X minus two is what we started with. So this is an accurate decomposition. But I mentioned that there are many correct ways that we could decompose functions. And let's take a look at some of the other strategies we could have taken. So another strategy we could use starting with our original function F of G of X. And remember this is just always going to be equal to the square root of X minus two. In this example, another strategy we could have taken is to set G of X equal to the entire thing. So if we had our individual function functions F of X and G FX, we could say that G FX is equal to the entire square root of X minus two. And in this case, F of X would just be equal to X. Because notice if we took this function right here and plugged it in for X, we would once again get square root of X minus two. So this might seem way too simple and a little bit ridiculous. But this is technically a completely, a completely correct answer also because if we recompose these functions, we get what we started with. So this is actually a correct decomposition as well. In fact, when you get really good at this, you can actually try doing some really crazy things. So let's say we start again with our original function F of G of X. And this is always going to be square root of X minus two for this example. But you could make it something ridiculous. So you could say that G of X is equal to the square root of X minus two minus. And I'm just going to come up with something here. So we'll say minus 1000. And in this case, we could say that F of X, the original function is equal to X plus 1000 because notice if you were to take this entire thing and plug it in there, the thousands would cancel and you just be left with square root X minus two. So this is another weird but perfectly OK strategy that you could take for decomposing these functions. So once you get good at this, you could actually get creative with the various decompositions that you do. And 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.

9

example

Decomposition of Functions Example 1

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So let's see how we can solve this problem in this problem. We are given the function H of X is equal to one over X squared plus three, X minus 10. And we're told that this function is a composition of two functions F and G and to express these functions such that H of X is equal to F of G of X. So basically what this problem is asking us is to find individual functions F of X and G of X that when composed would give us this function H of X. So to figure this out, I'm actually going to write this H of X as a composition of these two functions. So we're going to say that F of G of X where G would be the inside function here is going to equal one over X squared plus three X minus 10. Now, in order to decompose this function, we need to figure out what F of X and G of X are going to be well to do this, I can think of some examples that I could do to split these into two functions. And one thing that I recognize is this is a fraction. And with a fraction, I can see that the denominator is the inside function and that the entire thing would be the outside function. So what we would end up having is G of X would equal everything in this denominator. So we'd have X where plus three X minus 10, this would be the G of X function. And then for F of X, we would say that F of X is everything that we did not set equal to G. So in this case, we would have the one over this entire thing which we said was G. So in this case, we would just say this is X. So this would be the functions for G and F of X. If we decompose this and notice if you were to take this function and plug it back into F of X, we would get this exact same thing that we had over here. So this right here would be the solution for decomposing this function. Now, there are other different things you could try with this. We discussed in the video on decompositions that you can get creative if you want to. But I think this is one of the most intuitive ways to decompose this function. So hopefully, you found this helpful. Thanks for watching. Let's move on.