In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use tra...

Question

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x) = 2 + log2 x

Accepted Answer

Hey, everyone. Here we are asked to graph the following function by graphing first, its parent function F of X is equal to log of X and performing transformation determine its vertical Asymptote domain. And range of the function G of X is equal to log of X plus four. Here we have four answer choice options, ABC and D, each of which state the vertical Asymptote domain and range as well as a graph including F of X G FX and the vertical Asymptote. So to begin solving this problem, we first want to focus on graphing the parent function, which if we recall is F of X is equal to log of X. So to graph this function, we first need to consider various X values. And we can make m by considering when X is equal to one. Well, to find when X is equal to one, we need to find F of one which is equal to log of one. And now simplifying log of one just gives us zero. So now we, we need to repeat this process for various X values and set up a table of values. So setting up a table of values for X and Y, we have, we can include the X values 0. and five. And then substituting in these X values, we get the Y values of negative 0.300 point 30.480 point six and 0.7. So now with our table of values, our next step is to begin graphing F of X by first plotting each of these ordered pairs. So now plotting these points on a graph, we get our six different points. And now with these six points, the next step is to just connect them with a smooth curve line. And so now we have graphed our parent function F of X which is shown as this smooth black line that approaches X is equal to zero. And we just can note that our function F of X is equal to log of X follows the form F of X is equal to log base B of X where again, the, this function approaches the Y axis or when X is equal to zero but does not touch the Y axis. So now we just want to consider our second function given which was G FX is equal to log of X plus four. And we see that this function follows the form G FX is equal to log base B of X plus C which is a transformation characterized by shifting the graph of F of X. Of log base B of er F of X is equal to log base B of X by upward by C units. So to graph G of X is equal to log of X plus four, we just need to shift the graph of F of X log base B up by four units. So shifting F of X up four units, we see we get a new graph where the top line, the top smooth line is now G of X and then the bottom line is just F of X which again was shifted up for units. So these arrows pointing upwards are just up, going upwards for units. So now with our new graph, we want to next identify the vertical Asymptote domain and range. So starting with our Asymptote, we can simply assess our graph here. And we see that for both G of X and F of X, both of these lines approach the Y axis but never touch the Y axis. So we can draw in a dash line for our vertical Asymptote here. So we have a vertical line along the Y axis or this is where X is equal to zero. So we see here that our vertical Asymptote, our vertical axum to is where X is equal to zero. And now moving on to identifying our domain, we can look at our X values. So we see we go from zero, our Asymptote. So we go from zero all the way to positive infinity. And now infinity is always excluded. So it gets closed by a parentheses and now zero is also excluded because that is where our vertical asom tode is. So zero also gets closed by a parentheses. And now for our range, we can look at our Y values in the graph and we see that we simply go from negative infinity to positive infinity since there are no restrictions on our Y values. So our range again is from negative infinity to positive infinity. So now with our graph of G FX F of X and our vertical as, as well as our domain and range identified, we see here that we are left with answer choice. A where again our vertical ascent to is X is equal to zero. Our domain is from zero to positive infinity and our range is from negative infinity to positive infinity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x) = 2 + log2 x

Key Concepts

Logarithmic Functions

Transformations of Functions

Asymptotes and Domain/Range

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