Graphing functions Use the guidelines of this section to make ...

Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = 3x/(x² - 1)

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Graph the function F of X is equal to 4 x divided by X2 minus 9. Awesome. So it appears for this particular problem, we're asked to graph this function of FFX. So in order to graph this function of FFX, let us first off, in order to help us solve this problem, let us focus our attention on finding the domain. So let us recall and note that the denominator states that X2 minus 9 is equal to 0. So when we rearrange this expression to isolate sulfur X, we will find that X is equal to 3, or X is equal to -3 as our final two answers for the domain, because once again, if we when we factor out X2 minus 9, we will find that X is equal to 3 and -3. So therefore, as a result, the domain will be all real numbers except X equals -3 and X equals 3. And once again, this is for our domain, which I'll put a capital D in quotes, so that's our domain. Now we need to solve for our symmetry, which I'll just denote as S. So we need to check our function at negative X. So when F of negative X, so once again, F of negative X is going to be equal to 4 of negative X or rather 4 multiplied by negative X because instead of saying F of X, we're saying X is actually going to be negative X. So when we plug in a value of negative X in for X, we will find that F of negative X is equal to 4 multiplied by negative X, divided by negative X to the power of 2, so negative X to the power 2 minus 9. So that will mean that negative F of X is going to be equal to -4X divided by X2 minus 9. Awesome. So, since F of negative X is equal to negative F of X, that will mean that the function is odd. Which means that it is symmetric with respect to the origins. So that means, so we can write the sounds, this is very important, so it's symmetric with respect to, which is known as WRT with respect to the origin. Which the origin point, as we should recall, is 0.0. So let's make a note of that. So this is at 0.0. Fantastic. So now, let us focus our attention on intercepts now, which I'll denote as I. So solving for our intercepts, that means we need to Start with our Y intercept. So let us set X is equal to 0. So when F of 0, so when X is equal to 0, we will find that it's equal to 4 multiplied by 0, divided by 0 to 2 minus 9, which is going to be equal to simply 0. Fantastic, so that is our. Beginning of so for our Y intercept. So once again, our Y intercept is gonna be at 0.0. So now we could solve for X intercepts. So when we do that, we need to take F of X is equal to 0. So that means we need to take 4 X divided by X of the power 2 minus 9 is equal to 0. So that will mean that 4 X is equal to 0, so X will be equal to 0 ultimately. So the only intercept yet again is going to be at 0.0. Which once again is at the origin point. Now we can solve for our asymptotes, which I'm going to denote as A. So we need to recall and note that vertical asymptotes occur where the denominator is 0 and the numerator is not 0, so we need to take X2 minus 9 is equal to 0. So that will mean when we isolate and solve for X, we will find that X is equal to -3 and X is equal to 3. And for our horizontal asymptotes, so let's make a note that this is our vertical asymptote which we'll call VA. Now to solve for our horizontal asymptote, which is HA. We need to note that in order to find a horizontal asymptote, this means that X has to approach positive or negative infinity. So plus or minus infinity, so as X approaches plus or minus infinity. Fantastic. So that will mean that F of X is Roughly equivalent to 4 X divided by X2, which is equal to 4 divided by X, which approaches 0. So that means that the horizontal ascent to will be Y equals 0. Fantastic. So now, We need to perform the first derivative to help us determine the critical points. So, with that said, and we need to find the derivative using the quotient rule. So we're calling and using the quotient rule. We can say that U is equal to 4 X and we could say that V is equal to X2 minus 9. So then we could say that U is equal to 4, and we could say that V is equal to 2 X. So that will mean that our first derivative, which states that F of X, where a prime denotes the first derivative, this prime symbol. So F of X is going to be equal to U multiplied by V minus U multiplied by V. Divided by V2, which is going to be equal to 4 multiplied by x 2 minus 9 minus 4 X multiplied by 2 X, all divided by X 2 minus 9 of 2. Fantastic. So X2 minus 9, all to the power 2, to be precise. So now we need to simplify the numerator, and when we simplify the numerator, we will find that F F X F X will be equal to -4, multiplied by X2 + 9. All divided by X2 minus 92. So I've gone ahead and skipped a few steps, but all you need to do is use some algebra to simplify the numerator down to its most simple form where you don't, or where you cannot simplify any more. And when you do, you'll get that F of X is equal to -4 multiplied by X2 + 9 for the numerator, divided by. X2 minus 9 all to the power 2 for the denominator. And we need to note that since X to the power 2 + 9 is going to be greater than 0 for all X, and the square in the denominator is always going to be positive except at X equals plus or minus 3, which are not within our domain. That will mean, without a doubt, we will be able to state that F of X is less than 0. For all allowed values of X, so that will mean that there are no points where F of X equals 0, because X2 + 9 equals 0 will have no real solutions. Therefore, without a doubt, there will be no critical points, so no critical points that you'll call CP, so there's going to be no critical points. It'll put an exclamation point. So no critical points from the setting, the first derivative equal to zero. The function is strictly decreasing on the interval of this specific domain. So now we need to perform the 2nd derivative and help us determine, and once we once we perform the second derivative, that will help us explore the concavity. So now we need to differentiate F of X is equal to -4 multiplied by X2 + 9, all divided by. X 2 minus 9 all 2 using the quotient rule, and when we do, we're calling and using the quotient rule, you will find that F of X is going to be equal to 8 X multiplied by X2 + 27, all divided by X2 minus 9 all 2. We now need to note that F of X would only be equal to 0 at X equals 0, thus X equals 0 will be a good candidate for an inflection point. And we need to recall and note that since the function has a vertical asymptotes at X equals -3 and X equals 3, we need to analyze the sign of F of X on the intervals negative infinity 3. 3.0 within parentheses. 0.3 in parentheses and 3 positive infinity. Fantastic. So, with that said, We now need to consider That we need to note that X to the power of 2, so we need to take X 2 plus 27 is greater than 0 for all X, such that the sign is to be determined by the factors 8 X and X2 minus 9 to the power 3. So for X is less than -3, we need to note that next, that this value of X is going to be negative. So I put a negative symbol, which is like a minus sign, and we need to note that. X2 minus 9 is going to be greater than 0, so thus the cube will be a positive value. So, therefore, As a result, F of X is going to be less than 0, and the function is going to be concave down, which I'm gonna denote as CD, so concave down. So now, for the interval, -3 is less than X and X is less than 0. We need to note that X is going to be negative, and we also need to know that X2 minus 9 is going to be less than 0, so that means that its cube will be negative. And because the negative is divided by a negative, which will give us a positive result, which will mean that F of X is going to be greater than 0, which will mean that for this particular interval, it's going to be concave up, which we'll call Cu. Fantastic. And for the interval, 0 is less than X and X is less than 3. We need to note that X is going to be positive in this case, and X to the power 2 minus 9 is less than 0 will give us a negative Q. Therefore, we can say that F of X is going to be less than 0 and it's going to be concave down. Fantastic. And for the interval, X is greater than 3. That will mean that X is going to have to be a positive value, and X2 minus 9 is going to be greater than 0, will mean that the cube is positive. So that will mean that F of X is going to be greater than 0, and it's going to be concave up. Fantastic. So therefore, there will be an inflection point where once again we need to recall and note that an inflection point states that there's that the concavity is going to change. So the inflection point where the concavity changes will occur at X equals 0, since the sign of F of X changes from positive to negative. So with all of this information that we collected, we can now sketch our graph. Using all the steps that we followed together, so when we take all the information that we collected together and we graph it, we will find that we will produce the following graph. Fantastic. So as you can see, We can see that we have, going from the far left of our graph working our way to the right, we have it decreasing concave down with our asymptotes. Represented by dotted lines, and we have our red asymptote, which is denoting X equals -3. And then we have our asyote. Once again, since we have to note that we have our vertical asymptotes at X equals -3 and X equals 3, which are denoted in red and purple dotted lines respectfully, and we have a cubic graph in the center between our two vertical asymptotes, which is decreasing, and it starts in the upper part of the quadrant being concave up, where in the bottom quadrant it's concave down. And then moving our way to the far right of our graph, we have it decreasing being concave up, and all of our curves are denoted in blue. And that's it, we've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 3x/(x² - 1)

Key Concepts

Function Behavior

Finding Asymptotes

Critical Points and Intervals

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