The figure shows the graph of f(x) = ln x. In Exercises 65–74, us...

Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.
h(x) = ln (2x)

Graph of f(x) = ln x showing key points and asymptotes for transformations.

h(x) = ln (2x)

Accepted Answer

Hey, everyone here we are asked to use the graph of F of X is equal to the natural log of X shown and perform transformation to graph G FX is equal to the natural log of four X. We also need to identify the domain range in the equation of the Asymptote. Here we have four answer choice options. ABC and D. Each of which display a graph including the function G FX as well as the vertical Asymptote. And each answer choice states the vertical Asymptote domain and range. So now moving on to solving our problem here, we first need to consider the given graph of F of X which again is equal to the natural log of X. And we see that from this graph of F of X, we get four ordered pairs. So the ordered pairs are 0.5 and negative 0.7. Then we have one and zero. Then our third point is five and 1.6 and our fourth point is 10 and 2.3. So if we recall the function that we are graphing here is G of X is equal to the natural log of four X. Well, we see that this function matches or follows the formula G FX is equal to the natural log of C X. And here we see that C comparing our formula to our function that C is equal to four, which is greater than one. So we know that the transformation is horizontal shrinking. So we know that to graph G of X is equal to the natural log of or X, the X coordinates of the ordered ordered pairs from our F of X graph must be divided by four. So we can set up a table of values. First, we have our X value values, then we will have our X divided by four values and then our Y values. So starting with our X values again, we have 0.515 and 10, then we need to divide each of these values by four. So we have 0.5 divided by four which simplifies to 0.125. Then from our next X value, we have one divided by four which simplify 20.25. Then we have five divided by four which simplifies to 1.25. In our last value, we now have 10 divided by four which has a decimal equivalent of 2.5. And now our Y values remain the same. So we have negative 0.701 point six and 2.3. So now we want to grab the points that we found for X divided by four and the Y values. So these are our new ordered pairs. So we have four new points and graphing these points on a graph, we see we have our four different points. And so the next step is to just draw a smooth line that crosses through all of these points to graph G of X. And so drawing in this line, we see we now have our graph four G of X, which if we recall is equal to the natural log of four X. And now we can move on to identifying our vertical atom toll. So our vert vertical Asymptote can be determined just by looking at our graph. And we see that this line of G FX approaches the Y axis but never crosses the Y axis. So we see that our vertical Asymptote is going to be a dash line on the Y axis. And so this tells us that our vertical Asymptote is when X is equal to zero. So we have a vertical Asymptote at X is equal to zero. Now, we can move on to identifying our domain again by just looking at our graph here, we look at our X values for the domain and we see we go from our vertical Asymptote at zero all the way to positive infinity. Now, infinity is always excluded. So it gets closed by a parentheses. And we also know that zero has to be excluded since this is where we have our Asymptote. So zero also gets closed by a parentheses. Now we can identify our range. Again, looking at our graph, we now look at the Y values and we see that we do not have any restrictions on our Y values. So we simply go from negative infinity to positive infinity and infinity is always excluded. So both terms get closed by parentheses. So now we have graphed our function G of X and included our vertical Asymptote and we also identified our domain and range. So with all of these pieces of information, moving back to our answer choice options, we see our answer choice matches answer choice C where we have again, a vertical Asymptote at X is equal to zero, a domain uh from zero to positive infinity and a range from negative infinity to positive infinity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.