Given the functions f(x)=x+3f(x)=x+3f(x)=x+3 and g(x)=x2g(x)= x^2g(x)=x2 find (f∘g)(2)(f∘g)(2)(f∘g)(2) and (g∘f)(2)(g∘f)(2)(g∘f)(2).

( f ∘ g ) ( 2 ) = 7 (f∘g)(2)=7 ( f ∘ g ) ( 2 ) = 7 ; ( g ∘ f ) ( 2 ) = 25 (g∘f)(2)=25 ( g ∘ f ) ( 2 ) = 25

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

Let's give this problem a try. So in this problem, we have 2 functions. We have f of x is equal to x plus 3 and g of x is equal to x squared. For part a of this problem, we're asked to find f of g of x or excuse me, f of g of 2, and then for part b, we're asked to find g of f of 2. So So let's see if we can start by figuring out part a. For part a, we have f of g of 2 where g is the inside function. Notice because g is the second letter we see, it goes on the inside. Now to do this, that recall from the video where we talked about evaluating compose functions, there are a couple different strategies we can take. One strategy is to find f of g of x first and then evaluate this at 2, and the other strategy is to work from the inside out. 1st, evaluate g of 2 and then evaluate f of g of 2. Now for this first example here for part a, I'm going to start by finding f of g of x first, and then we're going to evaluate it at 2. So let's try this strategy. So what I'm gonna do is take g of x, which we have x squared, and plug it into f of x. If I replace x with x squared, this function f of g of x is gonna become x squared plus 3. Notice how the x just became x squared. Now from here, I can find f of g of 2. To find f of g of 2, we just need to take our x and replace it with 2. So So we're going to have 2 squared plus 3. 2 squared is 4 and 4 plus 3 is 7. So that means that f of g of 2 when evaluating this is going to come out to 7. So this is one of the methods we can use for evaluating composed functions, but let's try this other method for the second part. So the second part asks us to find g of f of 2, and rather than composing the function first and finding g of f of x, instead, we're going to find f of 2 first and then plug our function f of 2 into g. So let's see what we get. Well, if I try f of 2 here, f of 2 is going to be taking this x and replacing it with a 2 in our function f of x. So f of 2 is the same thing as 2 plus 3, and 2 plus 3 is equal to 5. So this is f of 2. Now at this point, we need to find g of f of 2, and to do this, well, we already got that f of 2 is 5, so in this case for our function, we're gonna look over to g and we're going to take this x here and replace it with 5, so we're going to have 5 squared and 5 squared is the same thing as 5 times 5, which is 25. So when we evaluate g of f of 2, we end up getting 25. So this is how you can evaluate composed functions. These were some more examples. Hope you found this video helpful, and let me know if you have any questions.