Log in RobertLLJones 8 years agoPosted 8 years ago. Direct link to RobertLLJones's post “could you still answer f(...” could you still answer f(g(x)) without knowing what "x" is? • (18 votes) Pranav Charvu 8 years agoPosted 8 years ago. Direct link to Pranav Charvu's post “Yes. You can literally pl...” Yes. You can literally plug in the whole equation for g(x) in for f(x). For example: f(x) = 2x + 1 and g(x) = 4/x Then to solve for f(g(x)), you would plug in g(x) (the whole formula), in to f(x) for x. So... f(g(x)) = 2(4/x) + 1 = 8/x + 1 This is just a simple example but you can do it with many more complicated formulas as well. Yes I know that these formulas are not as complicated as many people doing Algebra 2 could do, but I was just giving another example... Hope this helps... :) (86 votes) Shawn Hsu 7 years agoPosted 7 years ago. Direct link to Shawn Hsu's post “can a function have more ...” can a function have more than one variable? like: f(a,b) or something • (28 votes) Evan Indge 7 years agoPosted 7 years ago. Direct link to Evan Indge's post “Absolutely! A function wi...” Absolutely! A function with more than one variable would be called a multivariable function. and that is actually the correct way of notating it! A function with 2 variables would make a 3D object, 3 would make a 4D object, etc. Great question! Nathan Burkhart 9 years agoPosted 9 years ago. Direct link to Nathan Burkhart's post “will i need this for real...” will i need this for real life. plzzz help i need to kno • (0 votes) Raphi K 9 years agoPosted 9 years ago. Direct link to Raphi K's post “Yes; say that I sell cars...” Yes; say that I sell cars. In the function f(x) = y, the input (or x) is the number of cars being made in the year and the output (or y) is the price one needs to sell a car for. And then I will need to calculate the least amount of cars I would have to sell in order to earn enough money to pay my employees. The price of the car in that year will be P and the number of cars I will need to sell will be S, so I will now have a new function called g(P) = S. (68 votes) BaChu a year agoPosted a year ago. Direct link to BaChu's post “*WHAT IS THIS, I'm totall...” WHAT IS THIS, I'm totally CONFUSED- • (12 votes) SW 6 months agoPosted 6 months ago. Direct link to SW's post “Hello Barret” Hello Barret (2 votes) Logeswaran SaCh 2 years agoPosted 2 years ago. Direct link to Logeswaran SaCh's post “Hey guysIs it necessary ...” Hey guys Kim Seidel 2 years agoPosted 2 years ago. Direct link to Kim Seidel's post “Yes. There are foundatio...” Yes. There are foundation skills in Algebra 2 that you need to know in precalculus. (11 votes) detuncdanie 8 years agoPosted 8 years ago. Direct link to detuncdanie's post “At 2:24 where did you get...” At • (6 votes) hanegasa 8 years agoPosted 8 years ago. Direct link to hanegasa's post “Sal is plugging in (-3) f...” Sal is plugging in (-3) for x in f(x). f(x) = x^2 - 1, so plugging in x makes f(x) = (-3)^2 - 1. He also gets the -3 from g(2) because g(2) is the input into the function f(x). (6 votes) pnraju11 a year agoPosted a year ago. Direct link to pnraju11's post “I have a question that is...” I have a question that is more general. I'm taking Precalculus next year, and was planning on studying in advance through Khan Academy. Has anyone taken Precalculus before? If so, does the curriculum in Khan Academy align with what is being taught at school? • (7 votes) rsun 10 months agoPosted 10 months ago. Direct link to rsun's post “Precalc Honors for me inc...” Precalc Honors for me included topics such as: (3 votes) Nechama 10 years agoPosted 10 years ago. Direct link to Nechama's post “is the chart and graph th...” is the chart and graph the same for each example? • (7 votes) Cara 8 years agoPosted 8 years ago. Direct link to Cara's post “(Ignore top please)On my...” (Ignore top please) ....Oh wait, does any real possible answer to x minus itself make a zero? And if so, • (4 votes) andrewp18 8 years agoPosted 8 years ago. Direct link to andrewp18's post “ƒ(x) = x(x - 5)(2x - 4)ƒ...” ƒ(x) = x(x - 5)(2x - 4) (5 votes) 24jkrueger 6 months agoPosted 6 months ago. Direct link to 24jkrueger's post “I have a few questions, w...” I have a few questions, why is it called class 12 math (India)? Is it like Indian math or something? Also I am in 8th grade, do you think I am too young to be learning this? Is this only for 12th graders? • (4 votes) Kim Seidel 6 months agoPosted 6 months ago. Direct link to Kim Seidel's post “KhanAcademy has multiple ...” KhanAcademy has multiple courses. They customize them to a target audience. However, they don't create new videos for each course. A course for the India curriculum likely uses this video as does the Pre-Calculus course for the US. (3 votes)Want to join the conversation?
For example:
(given f(x) = 2x+3 and g(x) = -x^2 + 5
( f o g)(x) = f (g(x))
= f (–x^2 + 5)
= 2(g(x)) + 3
= 2(–x^2 + 5) + 3
= –2x^2 + 10 + 3
= –2x^2 + 13
However, it's too much unnecessary work for me to have those two separated because P, the price one needs to sell a car for, equals my definition of y, which equals f(x). So instead of going through two functions, I can now use g(f(x)) to find out how many cars I need to sell in the least in each year.
Is it necessary to complete Algebra 2 before staring Precalculus ?
2:24
Graphing linear, exponential, quadratics, cubics, quartics, stuff with higher degrees, logs, natural logs, powers of e.
Solving inequalities, factoring, difference of squares, difference quotient, stretch/shrink of graphs, vertical/horizontal shifts of graphs, reflections of graphs, horizontal/vertical asymptotes, properties of logs, solving logs, unit circle, radians, degrees, linear velocity, angular velocity, graphing trig functions, angle of depression/elevation, domain/range of trig functions, solving triangles, rule of sines, rule of cosines, equations/graphing/rules of: roses, limacons, cardioids, parabolas, ellipses, hyperbolas, and parametrics. Limits, sigma notation, geometric and arithmetic, half life, apert formula, a=pe(1+r/n)^nt thing, explicit/recursive.
This isn't a comprehensive list, since I don't recall everything, and also what I learn may not be what you learn, but this should give you a general idea.
On my study guide, it says that F(x)= 8 =x(x-5)(2x-4). I know that an easy guess for x is 1, and if you write it out you can confirm it without strenuous arithmetic. but I don't understand how it goes from F(x) is 8, and 8=F(1), to saying that (x-1) is a factor of the equation 0 = 2x^3 - 14x^2 + 20x - 8
(2x^3 - 14x^2 + 20x is the standard form of x(x-5)(2x-4))
Also, is there an easier way to find a possible x (such as the 1 in the above equations) than guessing?
ƒ(1) = 8
That means when you plug in 1 for "x" in the above expression, you will get 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). We know that ƒ(1) = 8. It follows that if we divide ƒ(x) by (x - 1), then our remainder is 8. We can make (x - 1) a factor of ƒ(x) if we add something to the function that will get rid of the remainder. Since the remainder is 8 and we want to get rid of that, we subtract 8 to get:
ƒ₁(x) = 2x³ - 14x² + 20x - 8
If we divide by (x - 1) our remainder is:
ƒ₁(1)
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. Therefore (x - 1) is indeed a factor of 2x³ - 14x² + 20x - 8.
Your second question asks if there is an easier way to solve the following equation:
x(x - 5)(2x - 4) = 8
Here is the systematic algebraic way to do it:
First expand the LHS:
2x³ - 14x² + 20x = 8
Subtract 8:
2x³ - 14x² + 20x - 8 = 0
We solve this equation by factoring it. 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. We can do this with synthetic division. This yields:
(x - 1)(2x² - 12x + 8) = 0
We can continue to search for roots by finding the roots of the quadratic:
2(x - 1)(x² - 6x + 4) = 0
(x - 1)(x² - 6x + 4) = 0
The quadratic is not factorable. The quadratic formula yields roots 3 ± √5.
Therefore the solutions to the equation:
x(x - 5)(2x - 4) = 8
are:
x ∈ {1, 3 ± √5}
Comment if you have questions.
Video transcript
Voiceover:So we have three different function definitions here. This is F of X in blue, herewe map between different values of T and what G of T would be. So you could use this asa definition of G of T. And here we map from X to H of X. So for example, when X is equal to three, H of X is equal to zero. When X is equal to one,H of X is equal to two. And actually let me number this one, two, three, just like that. Now what I want to do inthis video is introduce you to the idea of composing functions. Now what does it meanto compose functions? Well that means to buildup a function by composing one function of other functions or I guess you could think of nesting them. What do I mean by that? Well, let's think aboutwhat it means to evaluate F of, not X, but we'regoing to evaluate F of, actually let's just startwith a little warm-up. Let's evaluate F of G of two. 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. Well it seems kind of daunting at first, if you're not veryfamiliar with the notation, but we just have toremember what a function is. A function is just a mapping from one set of numbers to another. So for example, whenwe're saying G of two, that means take the number two, input it into the function G andthen you're going to get an output which we aregoing to call G of two. Now we're going to usethat output, G of two, and then input it into the function F. So we're going to inputit into the function F, and what we're going to get is F of the thing that weinputted, F of G of two. So let's just take it step by step. What is G of two? Well when T is equal to two,G of two is negative three. So when I put negative threeinto F, what am I going to get? Well, I'm going to get negativethree squared minus one, which is nine minus one whichis going to be equal to eight. So this right over here is equal to eight. F of G of two is goingto be equal to eight. Now, what would, usingthis same exact logic, what would F of H of two be? And once again, I encourageyou to pause the video and think about it on your own. Well let's think aboutit this way, instead of doing it using this littlediagram, here everywhere you see the input is X,whatever the input is you square it and minus one. Here the input is H oftwo, and so we're going to take the input, which is Hof two, and we're going to square it and we're going to subtract one. So F of H of two is H oftwo squared minus one. Now what is H of two? When X is equal to two, H of two is one. So H of two is one, so sinceH 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. We could have done itwith the diagram way, we could have said, heywe're going to input two into H, if you inputtwo into H you get one, so that is H of two right over here. So that is H of two, and thenwe're going to input that into F, which is goingto give us F of one. F of one is one squaredminus one, which is zero. So this right over here is F of H of two. H of two is the inputinto F, so the output is going to be F of ourinput, F of H of two. Now we can go even further,let's do a composite. Let's compose three ofthese functions together. So let's take, and I'm doingthis on the fly a little bit, so I hope it's a goodresult, G of F of two, and let me just thinkabout this for one second. So that's going to be G of F of two, and let's take H of G ofF of two, just for fun. Now we're really doinga triple composition. So there's a bunch ofways we could do this. One way is to just try toevaluate what is F of two. Well F of two is going to beequal to two squared minus one. It's going to be four minus one or three. So this is going to be equal to three. Now what is G of three? G of three is when T is equalto three, G of three is four. So G of three, this whole thing, is four. F of two is three, three of G is four. What is H of four? Well we can just look backto our original graph here. When X is four, H of four is negative one. So H of G of F of two, isjust equal to negative one. So hopefully this yousomewhat familiar with how to evaluate the composition of functions.
