We're saying that hey, the x value, the x solution here is roughly 1.66.\): Graph of a parabola with its vertex at \((-2, -3)\). Let's make two functions, and then let's graph themĪnd see where they intersect And the x value where they intersect well that would be a solution to that equation, and that's exactly what Or reframe your problem in a way that makes it easier to solve. So the whole point here is, is that, even when it's algebraically difficult to solve something you could set up or restate your problem And if you were to actuallyįind the exact solution, you would actually find So this looks like it's about, I'll say this is approximately 1.66. So this is 1.6 and then it looks like it's about two thirds of Value, right over here it looks like it is happening at around, let's see this is 1.5, andĮach of these is a tenth. But even if we're trying to approximate, just looking at the graph. In fact you can even scroll over this and it'll even tell you You can keep zooming in in order to get to get a Get an exact answer, but even at this level of zoom and on a tool like Desmos And then in this yellowish color I have y is equal to g of x which is equal to one over the cube root of x, and we can see where they intersect. Y is equal to f of x which is equal to two to the Have two, we have f of x, or I could even say this is Our two sides of our equation but now we've expressedĮach of them as a function. So I graphed this ahead of time on Desmos. Point of intersection is, and so let's do that. Or we could go to a site like Desmos and graph it and at least try to approximate what the And so what we could do is we could go to a graphing calculator, Or another way to thinkĪbout it is they're going to intersect at an x value, where these two expressionsĪre equal to each other. X squared minus three is giving you the same y as Because where they intersect that means two to the And then you could graph each of these and then you could see And you had another that was y is equal to one over the cube root of x. Was y is equal to two x, two to the x squared What if I had one function or one equation, that And the way that we can do that is we can say hey, well Have things like computers, we can graph things and they can at least get us really close to knowing what the solution is. Gave you this equation is to appreciate that some equations are not so easy to solve algebraically. But once again not havin'Īn easy time solve this. Three x squared plus nine is equal to log base two of x. I could try to take logīase two of both sides and I'd get negative Negative three is just 1, so that's just going to be equal to x. Terms times negative three, is equal to x to the negativeġ/3 to the negative three. Negative three x squared plus nine power. And so then I would get, if I raise something to an exponent, then raise that to an exponent, I could just multiply the exponents. Maybe I can simplify itīy raising both sides to the negative three power. Is one over x to the 1/3, so this is x to the negative 1/3 power. To the x squared minus three is equal to x to the, I could rewrite this, this The way that I would at leastĪttempt to tackle it is, you would say this is two Wanted to solve this equation, two to the x squared minus three power is equal to one over the cube root of x. Not only you, but other people that may read this comment. What this means is that, for every x in the x-axis, we input that into the function we're graphing, and the result is going to be the y-coordinate of that x. We can also always graph a function, denoting the y-axis as f(x). By consequence, f(x) = +/-√x is not a function, because the square root of 4 can be either 2 or -2. f(x)=2x can never give different results for the same x. What this means is, if I input a number, the answer should always be only that one output. Things that are important about functions:Ī function can only be a function if it has only one output per input. What this function tells us is to square any number that we input into the function. The input is simply the x I chose (in this case 2) and the output was what the function spat out (in this case, 4).Īnd we can do more complicated functions, like: That means that, when we input any x into f(x), our answer is going to be 2 times that x. We can represent that as f(x) = 2x (reads "f of x equals two x"). Say we have a machine (function) that takes any number and doubles it. In this case, the person's name is the input, the mother's name is the output and the machine is the function. So, if I input myself, I would get as an answer my mother. * Give the name of the biological mother of that person Imagine you have a machine (call that a function) that, when fed with something, spits something else out.
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