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If f(x) =x2 _ 9x + 6 and g(x) = Tx, find (g 0 f)(4).(g 0 fJ(4) = |...

Question

If f(x) =x2 _ 9x + 6 and g(x) = Tx, find (g 0 f)(4).(g 0 fJ(4) = |

If f(x) =x2 _ 9x + 6 and g(x) = Tx, find (g 0 f)(4). (g 0 fJ(4) = |



Answers

For each pair of functions $f$ and $g,$ find all values of a for which $f(a)=g(a)$. $$f(x)=\frac{12}{x^{2}-6 x+9}$$, $$g(x)=\frac{4}{x-3}+\frac{2 x}{x-3}$$

Now he'll just considered a given function. That is F of X is given to us as minus four x plus seven. And there's another function that is geophysics, which is given to us as X square plus nine X minus two. What do you have to data mine over here? Simply a five of six minus J off six. We have to just find out the value of this given expression. Were you fine? So now to do that, what I'm going to do is I'm going to find out ff six first and then I find out g of six. And then I just subtract with the values so f f six would be founded by by what? Simply I'll be just substituting X s six over here, right? So that would be equal to minus 24 plus seven and minus 24 plus seven gives you more simply minus 17. Similar g Just do the same thing for G of six. G f six will be founded by the sub shooting Xs six in G of six g of x worry. So that would be X square. That is 36 last 19 6 give you 54 minus two. So 36 plus 54 minus two gives you 88. Now F of six minus three of six will be equal to minus 17, minus 88. And that is equal to minus 100. And right, so this is the required answer.

According to the inclusion we have function affects is to the ex wrestler and they have function. F Best lead X This is a photo six minus one, divided like two x This we can also write as it's fixed blessed again, but this is required to six minus expert toe X divided by Do not affect we already know. So we can I d Exit acquittal six minus extroverted like minus ethics. So this is six minus X divided by two minus three x plus one Who will open the record? This is six minus six, divided by two minor minus Love the six miners when you sleep on mine African Dean Kamen This is excavated by two last year Legal right, this is flight minus. This is X plus six X divided by two. So that means we consider this is equal to five minus seven divided by toe digital many off the IT function

In this problem, we are asked to solve seven different problems revolving around these two functions. F N g. The first function. The first question is asking, When does the function f of X which, by the way, is over here on the right negative X squared minus X plus one. When does that function equals zero and you need to know what that means. Well, F of X is the output values of the function f of X have of exit the output value X is the input value. And when X is the input value, When does the red underlying equals zero. When does this equals zero? And if I were to look at it in that way, now it's a solvable problem. So what exactly does that mean? I'm asking, What are the output values when the output values are equal to zero, where the input values using a graphing utility will help us out with that? Because the output values being the Y axis, the output values are zero when it is at the X axis. So looking here, when the output values are zero, that's right here, and that's everything across these the X axis and here I have F f of X already. So let's take a look. When does this function cross the X axis? It crosses that negative two and positive one. So when X equals negative, too, or when X equals one. And those were the two values that make the function f of X equal to zero. I'm gonna ask the same question for G of X now Windows G of X equals zero. And again, I have a parable here and I want to know what are the input values for when this green parabola is equal to zero and let's take a look at that green parabola. There it is G of X negative X squared plus X plus six. It crosses the X axis at two locations as well. Negative two and positive three. So at X equals negative two or at X equals negative positive. Three. Those were the two values for Win X. When G of X is equal to zero questions see windows f of X, equal G of X. And what that means is, when do these two parabolas cross each other? Because they have the exact same a value they have. They are going to be translations of each other. They're gonna have the exact same shape, and if they cross, they'll cross only once. So let's take a look at that. And lo and behold, they cross that negative, too, which you could have reduced from the fact that they both have the X intercept of negative to as well. So they equal each other at X equals negative, too. Question D win is F of X greater than zero. What that means is, when is this above win? Is the red parabola above the X axis? Because, remember, when the function equals zero, that's the X axis. So when is the function above the X axis? Let's take a look at the graph. It's above the X axis in between Negative two and positive one. That's gonna be negative, too, is less than X, which is less than one in between. My two X intercepts is when this function this upside down function is above zero. When is G of X less than or equal to zero taking Look that over here on the graph we have between negative two and three, it's above. That means anything to the left of negative too, is below the X axis. Anything to the right of negative of positive. Three. His above The X axis is below the X axis, so that means it's going to be X is X will be less than or equal to negative. Two or X will be greater than or equal to three everything to the right of three and everything to the left of negative, too. Question F When is f of X greater than G of X? And here again, I'm asking where the X values for when this output values are satisfied and because F and G are the upward values referring to the Y values. I want to know winner the Y values of f above the why values of G and that's a vertical concept. So when is the red line above the green line? So let's take a look at our proble. The red line is above the green line to the left of negative too. So everything to the left of negative too f of X will be greater than G of X. I don't use the equal to symbol here, and we put in here really quickly if I were to do this. This was this would be win F N g are equal to each other such as there. But this I'm not asking for equal to. I'm asking for above someone. Go ahead and get rid of that by using the undue up here. And last but not least, question G is asking When is f of X greater than or equal to one? When is it above or equal to one? So let's go ahead and turn on Y equals one. And there it is above, remember, above means greater than in this situation means above. Okay, so it's above between negative 1.618 and 0.618 That's gonna be negative. 1.618 is less than or equal to X, which is in less than or equal to 0.61 eight. And that is how you do this problem. Thank you very much. Have a good day

Okay, so we have these two functions F of X and G. F X. And what we're gonna do is we're gonna find a couple of composite functions using these two. First we're gonna find F of G of X at X is equal to four. So we just plug in G. Of X for every value of X and F of X. So this is just gonna be the the absolute value of one over X squared plus nine. And if we plug in X is equal to four, we get the absolute value of 1/16 plus nine which is equal to, I can't remember the absolute value, the absolute value of 1/25 which is just equal to 1/25 for part B. We're gonna find gff and a fine G F At X is equal to two. So here we just plug in F. Of X for each value of X and G of X. So we have this is equal to one over the absolute value of X squared plus nine. And the absolute value of X squared is just equal to X squared. Since we know that if we square anything, no matter if it's positive number or a negative number, we're going to get a positive number. So we can say this is equal to one over X squared plus nine. And if we plug into we get 1/4 plus nine, sequel to 1/13 for part to see what we're gonna do is we're gonna find F. F. F of X. F is equal or at X is equal to one. F. F. F. Of one. So this is just equal to the absolute value of the absolute value of X. And in this case X is equal to one. So we have the absolute value of the absolute value of one. Well the absolute value of one is one and the absolute value of that is again one. So this is just equal to one for part D. We're going to find composite function G. Of G at X is equal to zero. So if we look at G F X again we're gonna have to plug in one over X squared plus nine into this X component here. So we're going to get one over one over x squared plus nine squared plus nine. And so if we plug in zero for X we get one over 1/0 plus nine square. Just 1/9 squared plus nine Which is then equal to one over Whenever 81 Plus nine. And so now if we plug this into a calculator, one divided by 81 plus nine and then we take one and divide that by that we get point 11 096. So the sequel 2.11096.


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