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3.7.25Let h(x) = f(g(x)) and p(x) g(f(x)): Use the table below to compute the following derivativesh'(4) =(Simplify your answer:)a. h'(4) '(2)3 "...

Question

3.7.25Let h(x) = f(g(x)) and p(x) g(f(x)): Use the table below to compute the following derivativesh'(4) =(Simplify your answer:)a. h'(4) '(2)3 "(x)2 33

3.7.25 Let h(x) = f(g(x)) and p(x) g(f(x)): Use the table below to compute the following derivatives h'(4) = (Simplify your answer:) a. h'(4) '(2) 3 "(x) 2 3 3



Answers

Use the given functions f and $g$ $$ \begin{aligned} &f(x)=-x^{2}+3\\ &g(x)=-3 x+3 \end{aligned} $$

This problem is asking us to solve it. Six different different questions, given our problems over here on the right. So we have a function f of x and F of X is equal to negative X squared plus three. And knowing what I know about functions, the X squared is makes this a problem. And as a reminder, when it comes a function notation X is my input value while f of X is my output value. Same thing here G of X So X is my input and G of X is my output. And so we're looking at two different situations here, Um, one with a problem, one with a linear line. And we're looking at all six of these problems. And when you look at these problems, we need to think to ourselves. What does it mean for F of X equals zero? Well, we know that X is our input value, but what is equaling zero right now the F of X is equaling zero, which means my output value is zero. We already know where output value. The question here is what is my input value when the easiest way to determine that is to look at a graph. So coming over here, I am looking for When is the output value equal to zero? Well, that's at the X axis. So let's take a look. Er at the graph. The graph is here. F of X equals X squared is going. Zoom this back into the main area and you'll see that it crosses the X axis at two locations. Negative 1.732 and positive 1.732 And again, the output value. The why value is zero at these two locations. So 1.732 is my answer, both positive and negative. So I can say X equals negative. 1.732 or X equals 1.73 to the same thing is being asked for G of X. Now for question, be for question be we're asking when is the input value. What is the impact value when the output value is equal to zero? So coming back over here again, I have the graph already made for us. Negative three X plus three and looking at how it's drawn here, it crosses the X axis at the 0.10 that's gonna be my answer. X equals one when the output value is zero question, See, when does f of X equal G of X and again? We're asking the output values here. So when do the Y value is equal? What are the X values? So our answer is gonna ask us the question. What are the X values when the output values are the same? Well, what that means is where they cross and they cross at two locations. They cross at the 20.3 and the cross at the 0.3 negative six. But again, this question is asking us what is the input values when this happened? Because f of X and G of X are output values. So we're asking what are the input values? When this happens, the input values are zero and three, so X equals zero or X equals three. Question D win is f of X greater than zero. Basically, win is the proble above the X axis. So looking at the graph, let's come back to these points. It is above the X axis. In between these two values, every point here has a positive y value. As we want to know what those are. It's in between negative 1.732 and below 1.732 So that means we can write this as a compound inequality. Negative 1.7 32 is less than X, which is then less than 1.73 to question. D is now bringing us back to G of X, and we want to know when is G of X less than recalls. Zero. So back to the graph lesson request. Zero. When is the green line below the X axis winner? The Y values negative with Y values or negative below the X axis. So everything to the right of this point 10 So what are my ex values? Everything to the right of one that means X. It's going to be greater than or equal to one. That means everything to the right of one. F is a question that's relating to F. N. G. Is asking us to find Win are When is F When are the why values of F greater than the Y values of G translation. When it comes to a graph, win is the red line or the f function above greater than the Green line. So let's take a look. Here it is. It is above think up and down. The red line is above in between the points 03 and three negative six. So everything in between here is when the red line is above the green line. That's three zero is less than X, which is less than three again. We're looking at the X values, and the last question is asking us. Win is F of X greater than or equal to one again. This is the Y value of one. So I've actually done this already. I've created 1/3 line, and we're trying to find When does it cross? When is F of X greater than or equal to that purple line greater than or equal to meaning above that purple line? It's greater than or equal to that purple line in between negative 1.414 and positive 1.414 That's gonna give us negative 1.41 for is less than or equal to X. Because remember, at negative 1.41 for the function, F actually touches the number one, which is then less than or equal to 1.414

This problem is asking us to solve for six different scenarios. The key to this is understanding function notation. Here we have X X is our input value on all of these and then we have these two functions F of X n g of X and F of X is the output values of the function F Whatever the function, F is half of X is the output or the y values. So what we're looking at over here is we have the output values for f of X is X squared minus one. We know that this is going to be some kind of parabola. Okay, On the other hand, G of X is another graph, and that graph is determined by this function over here. And it's three x plus three. And it's so that's some kind of linear line, and we're asking to determine what are the output values. What input values will give us the satisfied requirements here. So, as an example for question A, it's asking us what are the input values that makes the output values zero. So let's come over here to the graph, and you'll notice that I already have the graph made well. What are the output values? The output values are the Y values. So what are the input values that gives us an output value of zero negative one and one. So coming over here, our answer to this problem is X is going to be one or negative one, and that is the answer to party. Moving on apart beef, Part B is asking Winner the output values of G equal to zero. So what output values? What input values give us an output value of zero? Well, coming over here to this three x plus three is what I'm looking for. Let's turn that graph off and turn this craft back on our out. The values are the wives axis. The Y axis is zero at the X axis. So when is the input values When will be in good values? Give us a zero output value at negative one. It's coming over here. The input value of X equal to negative one is our answer here. Part C. When does f of X equal G of X? What input values gives us this answer. So let's take a look. Let's turn both of these on and find out when does f of X n g of X equal? Let's go ahead and zoom out so we see both grafts and there we go and we have two graphs here. They meet here and here, and the question is, what are the input value? What are the X values? Where these air True, these are equal at eight of one and positive for. So the answer to this is X will equal negative one or four party is asking. When is the function f above the X axis? This right here, zero is the X axis. Let's find out what that's when that happens. So looking at only the green graph is above the X axis before negative one. And after a positive one. X is less than negative. One or X is greater than positive one. When his G of X greater than or less than or equal to zero. When is the output value of G less than or equal to zero? So let's look at the Purple line is less than or equal to zero before negative one. So before negative one, it's below the X axis, so that's going to give us a value of X is less than or equal to negative one. Why equal to? Because it is equal at negative? One appears well. So we got. We can't forget the equal to part. Last but not least, win is the function f above G of X. This is an interesting question. When is the function f above the function? G and I always like to think in terms of up and down. So let's take a look at this graph. When is the green above the purple? Well, everything's little left of this point and everything to the right of that point here in between negative one and four in this region, right here f of X. The green parabola is below the purple line. So it is above when X is less than or equal to negative one, or when X is greater than four and let me make a quick correction. It is less than negative ones. They equal at negative one, so it cannot be the equal at negative one, so it cannot be above the greens

This question is asking us to solve seven different problems, given a certain parameter of two functions f of X and G of X. And we were asking the question, What is happening between these functions and these results? Okay, so in this first question, A when F of X is equal to zero understanding function notation is important here. F of X is my output value. So I'm asking when is my output value zero and what is my output value? X squared minus one? I know from my experience that this is a problem. So I'm asking when does that parabola across the X axis? And I'm asking, What is the x value that satisfies this? So let's take a look at our problem. I've already have a graft over here, and here's my red parabola, and I see that it crosses the X axis at two locations. Negative one and positive one. It's my answer is going to be X equals Negative one or X will equal positive one, and then I'm going to do the same thing for question. Be where us asking When is G of X three X plus three equal to zero or windows across the X axis. So let's go and take a look at what that graph looks like. We have three X plus three here It crosses the X axis here, a negative one. So our answer is going to be negative. One X equals negative one X equals negative. One Questions, see is asking us the question. When is F of X? When are the output values of F of X and winner the output values of G of X The same? And we're asking, what is the X value? Because that's what they have in common. Let's take a look at the graph and find where they intersect. They intersect that negative one and positive for So the two x values that I'm looking for are negative. One and or X will also equal positive for questioned the when his f of X When is the red proble greater than the X axis? When is it above? That's how I read greater than when is it above? So taking a look This right here the red problem it's above when it's to the left of negative one and to the rights of positive one. The left is the inequality. Less than X is less than negative one or when X is above negative. One X is greater than negative one. Sorry, positive. One Question E is asking us when his g of X less than or equal to zero. So come back here to the graph and looking out. Where does it equal? It equals that negative one. When is it less than zero? When is it below the X axis to the left of negative one? So that's going to be X is less than or equal to negative. One question. F win his f of x above g of X, And so we want to look at the graph to determine that at the vex is above when is the red line above the green line to the left of negative one, and then it happens again to the right of positive fourth. So that's X is less than negative. One or X is greater than four. And last but not least, the last question is when is F of X greater than or equal to one? So let's take a look at our graph really quickly. And when it was zero, that was the X axis, but one is the output value of one is the Y value here, and I've already made a graph of y equals one, and you'll see that across of the exact the this particular purple line at these two values. Negative 1.414 and positive 1.414 But the question is asking us when is F of X greater than or equal to one? So when is it above this Purple Line or equal to this purple line? Well to the left of this value negative 1.414 and to the right of this value 1.414 that means X is less than or equal to negative. 1.414 or X is greater than 1.414

So we have a function here, G of X 0 to 2 X squared minus three. Um We want to find the derivative formula as well as confirmed the answer with the quotient rule. So whenever you have a sort of um exponents for a ah function, let's say F of X is equal to X to the N plus C. Then the derivative of that function would be and in front And then N -1 at the top plus C. So we can use this formula for G fx So G prime of X would be putting that two in front, so you're gonna do to and you've got to multiply that other to those initially in front And now your ex is raised to one And three is our constant. Um So this is actually not here, this is wrong. Um So three technically is not going to be there when we do derivatives. So we're just left with two X. I'm sorry, four x for ex for the answer now to confirm this, ah They want us to use the quotient rule. So the quotient rule states that if we have two functions or if we have a function on the top and bottom, then the quotient rule, let's say uh h of X is equal to um I don't want to use G let's use ah ze ev X all over uh cuba or Yeah, cubic. So if your function looks like this, then the quotient rule states that you take the derivative of the first function multiply that by the bottom function minus the river of the bottom function and multiply that by the top function all over. Bottom functions squared. Uh I don't like that. Let's do this. So I mean we just have a numerator in this case but we could we could do it. So G prime of X. So the top function derivative is a forex. Bottom function is one minus. Bottom function derivative will be zero times. Top function was just two X squared minus three and that's all over one squared. So the zero goes away and we're just left with for ex. So just to recap the formula for any exponents, any variable to an exponent, you put the and in front -1 from the end and then the cold fisher and the constant goes away. So that's why we got four x. And then we confirmed it using the course rule.


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