The figure shows the graph of f(x) = log x. In Exercises 59–64, u...

Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, 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.
g(x) = 1-log x

Graph of f(x) = log x with points and asymptotes for transformations.

g(x) = 1-log x

Accepted Answer

Hey, everyone here, we are asked to use the graph of F of X is equal to log of X shown and perform transformation so that we can graph G of X is equal to four minus log of X. Now, we also need to identify the domain range. And the equation of the Asymptote here we have four answer choice options. ABC and D each which display a graph that includes G FX and the vertical Asymptote. And each solution also states the vertical Asymptote domain and range. So to begin solving this problem, we first need to consider our given graph of F of X and we need to first identify each of the given ordered pairs in the graph. So first we have 0.1 and negative one. Then we have one and zero, then we have seven and 0.85. And lastly, we have 10 and one. So now that we have identified the ordered pairs of F of X, we want to recall the function that we are trying to graph, which is G of X and we had G FX is equal to four minus log of X. Well, for now we want to ignore the four in the function. So we're really just considering G of X is equal to negative log of X. And we see that this function here negative log of X follows the form of G of X is equal to negative log base B of X. And this function is characterized by the reflection of the graph F of X is equal to log base B FX about the X axis. So to reflect the graph about the X axis, we need to negate the Y coordinates of the ordered pairs that we identified. So our first ordered pair of 0.1 and negative one becomes 0.1 and positive one. Then we have one and zero, this remains the same. Then we have seven and negative 0.85. And lastly, we have 10 and negative one. So now we want to consider our original G of X function here which had this four minus log of X. So we want to consider the four that we ignored earlier. And this four tells us that the Y coordinate of the ordered pairs have to be added by four. Or in other words, the graph should be shifted by four units upwards. So our next step is to just add four to each of our Y coordinates. So now we have 0.1 and then we have one plus four for our Y coordinate. Then we have one and zero plus four for our second ordered pair, then we have seven and negative 0.85 plus four. Then we have 10 and negative one plus four. And now just simplifying each of these ordered pairs, we have 0.1 and five, then we have one and four, then we have seven and 3.15. And lastly, we have 10 and three. So now with this new set of ordered pairs, we can go ahead and plot them all on a graph. So doing so we now have this graph of our four points for G FX. And so now the next step after plotting these four points is to connect the four points. And so connecting these points, we get a smooth curve that represents our function G of X and G of X was equal to four minus log of X. So now we have graphed our function G of X. But the next step is to identify and draw in our vertical Asymptote and identify the domain and range as well. So moving on to the vertical ay to while we can find this by simply assessing the graph we have. So we see that, that the graph approaches what X is equal to zero but never crosses it. So our vertical Asymptote is at where X is equal to zero. So we can just draw this dash line in to represent our vertical Asymptote. And now just restating our vertical Asymptote, our vertical Asom tote is at X is equal to zero. And now continuing to assess our graph, we can identify our domain. And so our domain, we can look at our X axis and we see we go from the vertical ASOM tote which is from zero all the way to positive infinity. And both of these terms are closed by parentheses since infinity is always excluded and zero cannot be included either since it is our vertical as. And now the range we can look at the Y values and we see that there are no restrictions and we simply have a range from negative infinity to positive infinity. So with this, we have identified our domain, our range in our vertical asym tool. And we have grabbed the function G of X along with the vertical ascent to. So with all of this information that we found, we see here that we are left with answer choice B where again our vertical ascent is at X is equal to zero. Our domain is from zero excluded to positive infinity and the 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.