In Exercises 36–38, begin by graphing f(x) = log2 x Then use tran...

Question

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

Accepted Answer

Welcome back. I am so glad you're here. We are given this logarithmic function, we're given F of X equals blog base three of X plus one. And we're asked to do several things here. We need to graph it using transformation from the base function. We need to identify its X intercept its vertical asom tote and its domain and range. OK. The first step is to graph it. I'm going to draw a rough sketch of a Cartesian plane here, a horizontal X axis, a vertical Y axis and they meet together at the origin in the middle. What is its base function? The base function would be the function without any shifts. So the base function here would be the log based three of X. We recall from previous lessons that the log base three of X is going to have an X intercept at 10. So approximately here as our Y values get larger, then our X values are going to get larger, very quickly. So that is going to curve this way very quickly. And as our Y values get smaller as they get more negative, and then our X values are going to get closer and closer to zero approaching zero because they're fractions but never quite reaching zero. So there is an Asymptote for the base function at X equals zero. So with transformations, what's the difference between our log base three of X base function and our log base three of X plus one, the difference is this plus one because that's happening directly to our X. This is going to be a horizontal shift and the shift that takes place is going to shift to the left because we're adding one, it's going to shift to the left one unit. So instead of remember our base function had an X intercept at 10, our transformed function is going to have an X intercept at 00. So it will still kind of curve pretty quickly this way. And now this time, instead of approaching zero because it's shifting to the left one, it is going to approach negative one. So it's going to have an Asymptote, a vertical asom tote at X equals negative one kind of draw dotted line for our as tote here. OK. So we have identified our X intercept that is at 00. We have identified our vertical asom tote that is at X equals negative one. And we've drawn our graph now we need to identify the domain and range. Well, the domain is going to be the X values that are possible because we have an Asymptote at X equals negative one. Our X values need to be greater than negative one. How about the range? All of the possible Y values? Well, the Y values could be anything from negative infinity to positive infinity. So the range is all real numbers. All right. So we have identified those. Now, we need to look at our answer choices to see which one matches up. And the clue to look for, to quickly rule out some things is often looking for intercepts. In this case, we'll look for 000 That changed color, didn't it? We'll look for 00. Looking at answer choice A for an X intercept. They have 10. So right off the bat, we know this one doesn't work for answer choice B they have an intercept at 10. That was our base function, not our transformed function. So B doesn't work. All right. Now let's scroll down. Look at C and D, I'm going to bring our graph with so we can take a look. Alright. C and D now C does have 00 for that X intercept. Great. Um C has got a vertical asom tote at X equals negative one. That is correct. The domain is that X has to be greater than negative one. That is correct. And the range is all real numbers. That is correct. And our graph looks pretty darn close to that. Now, let's look at D what is different about D they've got the 00 for the X intercept. That's good. But, oh, there it is the vertical asom tote they listed as X equals one instead of negative one. So that's why D doesn't work. And our answer choice here is answer choice. C well done. We'll catch you on the next one.

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

Key Concepts

Logarithmic Functions

Transformations of Functions

Domain and Range

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