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6. (12 marks total) Describe two methods that can be used t0 graph function and its inverse: (2 marks) Graph the equation y and its inverse on the same grid, along ...

Question

6. (12 marks total) Describe two methods that can be used t0 graph function and its inverse: (2 marks) Graph the equation y and its inverse on the same grid, along with the mirror line.y = €. (2 marks) List the special features of the exponential function and its inverse. Include the domain, range, intercepts, and asymptotes. (6 marks) d. Is the inverse function? Why or why not? (2 marks)7 . (12 marks total) Graph y 5" and y on the same grid. (2 marks) equation y has been transformed from

6. (12 marks total) Describe two methods that can be used t0 graph function and its inverse: (2 marks) Graph the equation y and its inverse on the same grid, along with the mirror line.y = €. (2 marks) List the special features of the exponential function and its inverse. Include the domain, range, intercepts, and asymptotes. (6 marks) d. Is the inverse function? Why or why not? (2 marks) 7 . (12 marks total) Graph y 5" and y on the same grid. (2 marks) equation y has been transformed from the parent function mark) Describe how the Determine the inverse of y (1 mark) d. Graph y and its inverse on the same grid (2 marks) how the transformed graph in part (a) of this question differs from the graph of the inverse in part (c) by Describe Include the domain; range, intercepts_ and asymptotes in your discussion. (8 marks) comparing the major features _



Answers

Functions such as the pairs in Exercises $69-72$ are called inverse functions, because the result of composition in both directions is the identity function. (Inverse functions will be discussed in detail in Section 5.1.) In a square viewing window, graph $y_{1}=\sqrt[3]{x-6}$ and $y_{2}=x^{3}+6,$ an example of a pair of inverse functions. Now graph $y_{3}=x .$ Describe how the graph of $y_{2}$ can be obtained from the graph of $y_{1},$ using the graph $y_{3}=x$ as a basis for your description.

So in part a of this question we're dealing with why equals two to the X. Power. And so Y equals two to the X. Power is an exponential growth function. Right? Its growth since my base is greater than one, It's passing through the .001 as well as the 0.1 comma two and then two comma four, et cetera. And we want to sketch are inverse. So let's think about that to do that. We're going to switch our role of X. And Y. For each of the points on the graph of y equals two to the X. So 01. That's going to become the 10.1 comma zero one, comma 2. That's going to become the point two comma one and two comma four. That's going to become the 40.4 comma two. And so looking at our inverse of why equals two to the X. Power. We're going to have a graph that looks very much like this right now. If I want to know what my inverse is. Well, the inverse of an exponential function is a log rhythmic function having the same base. And so my inverse in this case which is hearing Green would just be Y equals the log based chew of X. Now on that inverse function, let's talk about several characteristics starting with the domain and range. Well, the domain of this log rhythmic function. My set of X values that I can plug in Is from 0 to Infinity. How about my range? Well my set of why values that can come out of a log arrhythmic function is all reels negative infinity to positive infinity, I believe this log arrhythmic function does have an X intercept that X intercept is right here at the 0.10. Right? So my X intercept is ones you in terms of a Y intercept. I believe there are none. I don't believe there are any Why intercepts here, right? Because this log rhythmic function has a vertical ascent toe along the Y axis. We in fact do have a vertical ascent toe and that vertical ascent tote this time is along the y axis. It is X equals right. And so that was part A. Now in part B we want to do the same thing, but for a different exponential function. So my exponential function now will be y equals 1/3 raised to the X power. This is going to be an exponential decay since my base is now less than one. Now it's still going to pass. Looking at the graph, they gave us through the .01. When my ex is negative one, I get a Y value of three. So this is the point negative 13. And when my ex is positive one, I have a Y value of one third. This is the 10.1 comma one third. And here is this exponential decay function. Okay, now I want to know what would my inverse function look like. Right? And we're doing this on the same set of axes. So we're just going to switch our role of X and Y. So instead of having the .01, I'm going to have the point one comma zero. Instead of having negative one comma three, I'm going to have three common negative one. So I'm gonna have three comma negative one, roughly speaking, that will be somewhere around here and instead of one comma third, I'm going to have a third comma one that's going to be here. And so connecting with a nice smooth curve. My inverse function in blue is going to look something like this. Okay, now here is the inverse. But what is the equation of that in verse? Well, again, the inverse of an exponential function is a log rhythmic function of the same base. And so my inverse function this time would be a log rhythmic function of the same base. It would be Y equals The Log Base 1 3rd of x. Yes, there's my inverse. Now, once I have that in verse again, I want to know some characteristics of this inverse. Starting with the domain and range. Well, the domain of a log rhythmic function. 02 infinity. Right? I can only take the log of something positive. Yeah, my range just like the log arrhythmic function we saw before, negative infinity. To positive infinity. And there's my range in terms of an X intercept, I still have an X intercept? At 10. Right? That was a point on the graph of that log rhythmic function. Why intercepts there were none, Right? Because again, I'm going to have a vertical ascent toe there along the Y axis. And so we do have a vertical ascent over here, just like the other log arrhythmic function. And that vertical ascent toe is at X equals zero. So hopefully, that made sense, and you have a great rest of the day.

Okay. So here um were given this graph and we see um that we have a function that goes from the well location um to the annual rainfall in inches. And were asked to state what the domain and the range. Well, we're going from right, we're going from is our domain all the possible inputs. So that would be our location and then the output is the annual rainfall in inches. So that would be our range. Okay. Um So the function is 1 to 1 where we can go back and forth between the location and the annual rainfall in inches. So to find the inverse function, we just interchange the elements in the domain with the elements in the range. Therefore we obtain the inverse inverse of F. So the inverse of F is going to be F in verse after the negative one means f inverse. So F inverse is from well from annual rainfall, right from annual rainfall in inches um to to the location. Right. So all we do is just switch the domain becomes the range and the range becomes the domain. Okay? So therefore what is the domain of the inverse function? Well the domain is just um The the output of the original function. So therefore the domain of the inverse function is well for 60 um zero. And then we have to 102 .01. And then we have 196. Um Yes .46 and then we have 191.02 were just coming down the outputs of the original function become the inputs of the inverse function. 182 0.87 So there's the set which becomes our domain of the inverse function. And the range is just the locations. So we have um mount way way, holly hawaii, um mantra via um Liberia, pago, pago, american, american, Samoa, um mala, main burma lie um and Papua new guinea as our range of the inverse function.

Okay. So from our given graph we see that we have our function F is from the title of our movies to the domestic growth in millions of dollars. So we have our five movies and then we go to the domestic growth in millions. So the domain is just the inputs of the domain of our original function is the title of the movies. And the range is the domestic growth in millions. Okay, so the function is 1-1. So to find the inverse, all we do is just interchange the elements in the domain with the elements in the range. Therefore we have f inverse. Right? Our inverse function hopes are inverse function. Um F inverse is just well from right. So from um let's just say, so if this is from this is called title A. And domestic growth in millions be right, then F inverse instead of from A to B. We're going from B to A. So we're going from domestic growth in millions to the title of the movie. Um is our inverse function. Therefore, the domain of the inverse function is just the domestic growth. So here is our domain of the inverse function. That would be the 461, 431 400 3, 57 3 30. And the range are just the titles. So Star Wars, Star Wars episode one, the phantom Menace, e. T, Jurassic park and Forest gump would be the range of the inverse function.

Okay, So here from our graph we see that we have a function F that goes from the age to the monthly cost of life insurance in dollars. Okay? So if you're 30 you're paying $7.9 48,040 cents and so on. So therefore the domain of the function is just where we're going. We're going from age Two monthly cost. So our domain um that domain represent age and the range represent the monthly cost of life insurance. So we therefore have a function that is 1-1. So to find the inverse, we just interchange the domain with the range. We just have our arrows basically going backwards going backwards for our for our inverse function. Therefore, the domain of the inverse function are these values? These would be our domain this $3 values. And the range, the range of our inverse function are just the ages. All right, take care.


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