Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordi...

Question

Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordinate system. Use the graphs to determine each function's domain and range.

Accepted Answer

Welcome back. I am so glad you're here. We are given two functions the F. Of X. Which is equal to three raised to the X and G of X. Which is equal to log base three of x. And we get to do a couple of things here. We are going to graph them and we are going to look for their domain and range. All right, well, let's start off. I am going to draw a rough sketch of a graph will have a vertical Y axis and a horizontal X axis. And they meet together the origin in the middle. How do we graph them? If you have F of X equals three rays to the X. Memories. That's great If you don't and you need help remembering exactly what's going on. You can always graph a few points. I like to choose something like 10 and negative. One for values to give me an idea of what's going on in the graph. So for X equals one. Then we're going to plug a one in where we've got three rays to the X. It'll be three raised to the first three, raised to the first equals three. Now for X equals zero. That's three raised to the zero. Well, anything raised to the zero is equal to one and for three raised to the negative first. That negative in the exponent tells us that we're dealing with a fraction. So that's 1/3. Well, it's 1/3 to the first which equals one third. So now let's graph these. Looking at our Cartesian plane. We start with the pair 13. So that will be going one to the right on our X axis and up three for our Y axis. So that's approximately here Now for 01. We started the origin and that we're moving nowhere for our X. value. And we're going up one for our y value. Now for negative 11 3rd will be going to the left one for our X value and we'll be going up one third for our Y value. So we've got something that looks like this so far we can connect these three points. But what's happening before and after? Well, as our X values had more and more toward negative infinity, that's where we have these negative exponents. So we're dealing with fractions. And so really what's happening is we have larger and larger numbers in our denominator as three gets raised to a larger and larger number. But because the denominators are getting so big are fractions are getting so small. So really these are approaching zero and we have what's called an assam. Tote at X equals zero. So that function line that F of X is going to approach zero as X is negative but it will never actually reach zero. How about X gets larger? Moving to the right toward positive infinity. Well that's just this exponents here. Three raised to the first race to the second to the third. So very quickly these numbers are going to get very large. And so this function continues up to the right at a steep pace. Alright, so we've got our F. Of X graft. Now let's work on G. F. X. Likewise, we can plot a few points, but with logs, I am going to actually assign some Y values and figure out the corresponding X values. And here is why we know that G. Of X equals log base three of X. When we have something in log form. So Y equals log with a base of X. And in this case we know that our B. Is equal to three. So this log form can be written in exponential form which will be be raised to the Y equals X. And in this case since B equals three, it'll be three raised to the Y equals X. And it's a lot easier to put in a value for Y. And then solve for X. So for this first one will do why equals one. So that's three raised to the first equals our X values. The three raised to the first is three for the next one, Y equals zero. So that's three raised to the zero. Anything raised to the zero is one. These look familiar and now fourth three raised to the negative first. Well that is our fraction 1/3 raised to the 1st and 1/3 raised to the first is one third. So now we pull out our points. Looking at our Cartesian plane for 31. We started the origin go three to the right and up one for our first point for 10. We start at the origin we move one to the right for our X. Value. And we don't move up or down at all for our Y value. Now for one third negative one we move to the right one third for our X. Value and move down one for our Y value. So approximately here we can connect these three dots. But what's happening before and after? Well as why gets more heads toward negative infinity. That's when we're dealing with this fraction. And so the Y values is they get larger and larger. That's in the denominator. The Y values get closer and closer toward negative infinity. So they will get larger and larger in the denominator which makes these fractions get smaller and smaller. So this will approach why equals zero but it will never quite reach zero. So we've got an awesome tote at Y equals zero this time. And now, how about as our Y values head toward positive infinity. Well, that's this exponent here. Three raised to the first to the second to the third. So these numbers are going to get very large for our X values very quickly and our Y values will increase more slowly. So this is what our graph looks like. Our G of X is here in blue and our F. Of X is in red. Now, how about the domain and range? So let's do the domain for our F of X. The domain is all of the possible X values. And for F of X they could be all real numbers. They could be anywhere from negative infinity to positive infinity. All real numbers for the domain. How about the range for affects well the ranges all the possible why values? And it can't be every Y value because we have an assam. Tote at X equals zero. It can never get to zero or below that. So our range are Y values for F of X need to be greater than zero. All right now, how about for the domain and range of our G of X. The domain again is all of the possible X values. So our domain for our blue line G of X will the X values? They have to be greater than zero. So X has to be greater than zero because of that assam. Tote at Y equals zero. How about our range or G. Of X. While our Y values, they could be all real numbers. They can be from negative infinity to positive infinity. All real numbers. Okay, so we have graphed these and we have identified our domain and range. Now we just need to find our matching answer choices. Um Sometimes what is helpful to look for our our intercepts. So for F of X we have a Y intercept at 01 and for G of X we have an X intercept at 10. So let's look for those intercepts on our answer choices. Looking at answer choice A. They've got for G. Of X. That 40 and for F of X that zero to know that does not work. Let's take a look at answer choice B. They have got G. Of X has a 10 intercept and F of X has a 01 intercept. Those are looking pretty good so far. Alright. And now about the domain of F of X. All real numbers. That is correct. The range of F of X. Y is greater than zero. Yes. The domain for G of X X is greater than zero. Yes. And the range for G of X. All real numbers B is looking pretty good. Let's just rule out those other ones in case we were missing something where they're very similar. All right, answer choice C. Well that has 10 for our G. Of X and it should be 10 for our G. Of X. That looks okay. But look, it's going in a different direction R. G of X went this way and they also have the F of X intercept at zero negative one instead of 01. So we know that C won't work and for answer choice D 300 negative two. No, those are not our intercept. So B does work. We look at this answer choice B. Well done, we'll catch you on the next one.

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Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordinate system. Use the graphs to determine each function's domain and range.