Graph f(x) = (1/2)^x and g(x) = log(1/2) x in the same rectangula...

Question

Graph f(x) = (1/2)^x and g(x) = log(1/2) x in the same rectangular coordinate system.

Accepted Answer

Hey, everyone here we are asked to graph the following equations, F of X is equal to the quantity of 1/4 raised to the power of X and G of X is equal to log base 1/4 of X. So here we have four answer choice options. Each displaying a graph with a different graph of the functions F of X and G of X. So to begin solving this problem, we can begin by first trying to graph the first given function F of X. So as we recall, F of X is equal to the quantity of 1/4 raised to the power of X. Well, having F of X be equal to this expression is the same as having Y is equal to the quantity of 1/4 raised to the power of X. So now to graph this equation here, we just want to consider values of X and the Y values that we get from X and then plot those points. So we can first consider when X is equal to negative two. Well, when X is negative two, we have Y is equal to the quantity of 1/4 raised to the power of negative two. While simplifying we just get 16. So when X is negative two, Y is 16. So now we just want to consider a few more points and make a table of values. So first for values of X, we can have negative two, negative 10 and one. And then substituting in these X values into our function or into our equation here, our Y values become 16 for negative two. And then we have 41 and 0.25. So now with this table of values here for X and Y, we can now plot these points on a graph and so plotting, we see all of our points, all of our, we have four points here. And so now the next step with these points is to just connect each of the points with a line to see our graph of our F of X function here. So connecting the lines, we get a, a this blue line connecting the points as a smooth curve that approaches X. So now we want to repeat this process with our other given function here. We had G of X is equal to log base 1/4 of X X. And so now we can again, let Y be equal to this expression here. So we have Y is equal to log base 1/4 of X. And now we just want to consider various X values like we did previously. But here, since logarithmic functions are defined only for when X is greater than zero, then we can only consider X values that are greater than zero. So we can begin when let's say X is equal to one. So substituting in one, we have Y is equal to log base 1/4 of one. And simplifying we just get Y is equal to zero. So now again, we want to consider a few more points in and create a table of values for X and Y. So for our values of X, we can choose 123 and four. And now substituting in these X values, we get the Y values zero, negative 0.5, negative 0.79 and negative one. So now with this table of values, we can plot each of these points on a graph. And so we get all of our four points plotted on a graph. And now the next step just like before is to connect these points as a with a smooth curved line. We see we have this smooth curved line that A approaches the Y axis. And so now we have graphed both of our functions individually. So the last step is to just connect or combine both of these final graphs. And so now combining, we have this graph with both of our functions. We see F of X is the line in red and then G of X is the line in green. And we see that both of these lines intersect at one point. And so now with our final graph here, we see from our answer choice options that our two graphs match answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Graph f(x) = (1/2)^x and g(x) = log(1/2) x in the same rectangular coordinate system.

Key Concepts

Exponential Functions

Logarithmic Functions

Graphing in Rectangular Coordinate Systems

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