now in this lesson we're going to talk about composite functions so let's say that f of x is equal to 3x minus 4 and that g of x let's say it's equal to x squared minus three what is f of g notice that this is different from f times g if you see a closed circle it's multiplication it's 3x minus 4 times x squared minus 3. but if you see like an open circle what it means is it's a composite function one function is inside of another g is inside of f so this is equivalent to f of g of x so how can we find f of g of x well let's replace g g is x squared minus three and notice that x squared minus three is inside of f so we're going to have to replace x with x squared minus 3. so this is going to be 3 times x squared minus 3 minus 4. so now let's distribute the 3. so that's 3x squared minus 9. minus four which is three x squared minus thirteen now what about g of f how can we find the value of g of f so this time f is inside of g so you can write it like this that's an equivalent expression let's replace f of x with three x minus four and now let's replace uh x with three x minus four so we're going to take this and plug it into here so instead of x squared minus 3 it's 3x minus 4 squared minus 3. 3x minus 4 squared is 3x minus 4 times 2x minus 4. so if we foil it 3x times 3x that's going to be 9x squared and then 3x times negative 4 that's negative 12x negative 4 times 3x is also negative 12x and then negative 4 times negative 4 is 16. negative 12 minus 12 is negative 24 16 minus 3 is 13. so this is the value of g of f of x now let's say that f of x let's say it's 5x plus 2. and now let's say that g of x is equal to x cubed minus four evaluate this function f of g of two how can we do so so first let's find the value of g of two so let's replace x with two this is going to be two raised to the third power minus four two times two times two three times is eight eight minus four is four so g of two is four so f of g of two is equivalent to f of four and now we can take four and plug it into that equation so it's five times four plus two five times four is twenty twenty plus two is twenty two so that's how you can find the value of f of g of two so try this one evaluate g of f let's say negative one so first evaluate f of negative one that's five times negative one plus two so make sure you're using this equation that's negative five plus two which is negative three so f of negative one we can replace it with negative three so now we're looking for g of negative three which we can plug it into here so it's negative three to the third power minus four which is negative 27 minus four and so the final answer is negative thirty one so that's how you can evaluate composite functions you