In Exercises 5–9, graph f and g in the same rectangular coordinat...

Question

In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = e^x and g(x) = 2e^(x/2)

Accepted Answer

Welcome back. I am so glad you're here. We are given two functions. Were given F of X equals E. Raised to the X. We are given G of X equals four times E. Raised to the X divided by two. And we have to do a few things here. We need to graph our F of X. So we'll draw a rough sketch of what that will look like with our horizontal X axis and our vertical Y axis. And then we need to apply transformations to that F of X in order to graph R. G of X. Then we are going to graph the assam totes of each of our two functions and then use the graph to determine the domain and range of both of the functions. So, we've got quite a few steps here. Let's first work on graphing our F of X equals E. Raised to the X function. All right. You might just know what the race to the X looks like. But if you don't you can plot a few points to get an idea of where might some intercepts be or just a gradual idea of what's going on with the graph. So, I'm just going to choose for X values. I'm going to choose 10 and negative one to give us an idea and we'll plot our y intercept that way. So, substituting one in for X. That's E raised to the one. Well, what's E E as we recall is a natural number that it is equal to about 2.718. And then it keeps going from there. But we're going to use 2.718 just for as much as I want to right here. So we're talking about 2.718 raised to the first power. Well 2.718. Race to the first power is 2.718. So that's one point. So we go one to the right on our X axis from the origin where the X. And the Y axis meet. We go one to the right for positive one for the X. Value. And then we go up about 2.7 on the Y axis. So that's approximately there. And next we're going to plot when we have E raised to the zero. Well, anything raised to the zero is going to equal one by definition. So our y intercept is where X equals zero. And we go up to one on the Y axis. Now let's plot e raised to the -1. Well when we have something raised to the negative one, that means that that number is going to flip down to the bottom of a fraction to get rid of the negative. So we'll have one over E. To the one which is as we know 2.718. So 1/2 0.718 on our graph is going to be really close to one half. So where X is negative one we go one to the left from the origin for our X is negative one. And then we go up to about one half for our Y value. So we have something looking approximately like that. And it's important to note where our intercept is. That makes it a lot easier when we're looking at our various answer choices to find the correct one. Okay, we've got F of X roughly drawn. I'm going to connect those three dots. We don't know exactly what happens after that. But we know it looks like that. Now for our G. Of X that is four times the E. Raised to the X divided by two. So what's different between our F. Of X and our G. Of X? What transformations are happening here? Well, the first one here that we notice is the E. Is being multiplied by four. That's not happening directly to our X. So that transformation means it's going to be vertical because it's not happening directly to the X. And because that four is greater than one, then it's going to be a stretch. So we have a vertical stretch going on here. What else is different? Well we see that it's not just E. To the X. It's E. To the X divided by two. So that divided by two or multiplied by one half That divided by two or multiplied by one half. That's happening directly to the X. So that's going to be a horizontal shift. And because that one half is between zero and one that is also going to be a stretch. So those are the transformations if stretching horizontally and vertical isn't really intuitive to you. You can always plot a few points. I'm just going to plot one point here to make the point. But you can plot a few points to see what that stretching or shrinking. It would have been shrinking looks like let's see what X. Or what our Y intercept is when X equals zero. So we have four times E raised to the 0/2. Okay, E raised to the 0/2 is E raised to the zero so four E raised to the zero and erase to the zero is one. So this is four times one. So here our Y intercept is going to be at 01. So we can oh excuse me, no, four times one, that's four. So our Y intercept will be at 04 for the G of X. And so it's going to be very similar but it's going to be up here more. So that will help us with our transformations now for the ASM totes. Well, what we notice is that when we have negative numbers for X values that we are dealing with fractions and if our X values get extremely large, like if it was E raised to the negative 1/1000 then that's going to be one over E to the 1/1000 and e Times itself 1000 times. It's going to be a huge number but because it's in a fraction that's going to be very very very close to zero. But fractions can become infinitely small. So it's never going to actually get to zero. So we have an assam tote right at the X axis right at where why Equals zero. We have an assam tote. And um the last thing we'll need to do is identify the domain and range of both functions. Okay, so looking at our little sketch of a graph here, we know that okay, it's going to be approaching zero as it goes to the left and then it'll be getting larger because we're dealing with exponents going to the right, we'll have something very similar happening with our G of X. So our domain that is all the possible X values for R. F of X and G of X. The domain here, X could be any real number. That's going to be all real numbers. But how about for our range? What are the possible why values will we know we have an acid tote at Y equals zero. So for both affects and G of X, why is going to have to be greater than zero? There's our range. Alright. We've answered all of these questions and now we need to figure out which of our four answer choices matches with this. So I want to look especially for the Y intercepts. So we want our affects our reddish oranges function line to have a Y intercept at 01. Okay. And so for this first one it does have a Y intercept is 01. And how about our G of X are blue function line that's going to have a Y intercept at 04. And this first one does. So that one is looking pretty good. Let's skim the other ones while we're looking. So does F of X have um AY intercept at here. It does not. So this one's out. Let's look at answer choice. C. Does affects have a Y intercept at 01, nope. It does not answer choice. C. Is out. And how about for our answer choice. D nope. There is nothing at 01 for our alphabet. X line. There is no intercept. So D. Is out. And now let's make sure the rest of the criteria is met for a We've got our F. Of X and our G of X are looking good. We have our ASM tote at Y equals zero and this looks great. So we choose. Oh and does it say our domain and range domain for F. And G. Set of all real numbers. Range for F. Y is greater than zero. Range for G. Y is greater than zero. This one works. Alright. So answer choice. A Well done. We'll catch you on the next one

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In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = e^x and g(x) = 2e^(x/2)

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