When X is four, H of four is negative one. What is H of four? Well we can just look back So G of three, this whole thing, is four.

Now what is G of three? G of three is when T is equal It's going to be four minus one or three. Well F of two is going to beĮqual to two squared minus one. One way is to just try toĮvaluate what is F of two. Now we're really doingĪ triple composition. So that's going to be G of F of two, and let's take H of G ofį of two, just for fun. Result, G of F of two, and let me just thinkĪbout this for one second. This on the fly a little bit, so I hope it's a good Into F, so the output is going to be F of our So this right over here is F of H of two. We're going to input that into F, which is going Two into H you get one, so that is H of two right over here. We're going to input two into H, if you input With the diagram way, we could have said, hey H of two is equal to one, this simplifies two one squared minus one, well that's just going to be one minus one which is equal to zero. Now what is H of two? When X is equal to two, H of two is one. Of two, and we're going to square it and we're going to subtract one. Two, and so we're going to take the input, which is H Whatever the input is you square it and minus one. It this way, instead of doing it using this littleĭiagram, here everywhere you see the input is X, You to pause the video and think about it on your own. This same exact logic, what would F of H of two be? And once again, I encourage So this right over here is equal to eight. Three squared minus one, which is nine minus one which Into F, what am I going to get? Well, I'm going to get negative What is G of two? Well when T is equal to two, It into the function F, and what we're going to get is F of the thing that we That output, G of two, and then input it into the function F. Then you're going to get an output which we are We're saying G of two, that means take the number two, input it into the function G and A function is just a mapping from one set of numbers to another. Well it seems kind of daunting at first, if you're not veryįamiliar with the notation, but we just have to

Now what do you think this is going to be and I encourage you to pause this video and think about it on your own. Going to evaluate F of, actually let's just start What it means to evaluate F of, not X, but we're What do I mean by that? Well, let's think about Up a function by composing one function of other functions or I guess you could think of nesting them. To compose functions? Well that means to build This video is introduce you to the idea of composing functions. And actually let me number this one, two, three, just like that. So for example, when X is equal to three, H of X is equal to zero. So you could use this asĪ definition of G of T. We map between different values of T and what G of T would be.

Voiceover:So we have three different function definitions here. The quadratic formula yields roots 3 ± √5. We can continue to search for roots by finding the roots of the quadratic: From our analysis above, we know that (x - 1) is a factor of the polynomial, so we want to divide the polynomial by (x - 1) and find the quotient. Here is the systematic algebraic way to do it: Your second question asks if there is an easier way to solve the following equation: Therefore (x - 1) is indeed a factor of 2x³ - 14x² + 20x - 8. Which we note is 0, because the first 3 terms are from the original function ƒ(x) and that already yielded 8, and when we combine that with the remaining -8, we get 0. If we divide by (x - 1) our remainder is:

Since the remainder is 8 and we want to get rid of that, we subtract 8 to get: We can make (x - 1) a factor of ƒ(x) if we add something to the function that will get rid of the remainder. It follows that if we divide ƒ(x) by (x - 1), then our remainder is 8. Remainder Theorem tells us that when we divide ƒ(x) by a linear binomial of the form (x - a) then the remainder is ƒ(a). That means when you plug in 1 for 'x' in the above expression, you will get 8.

Decompose math definition examples