Graph y= 2^x and x = 2^y in the same rectangular coordinate syste...

Question

Graph y= 2^x and x = 2^y in the same rectangular coordinate system.

Accepted Answer

Hey, everyone here we are asked to graph the following functions in the same Cartesian coordinate system and find their point of intersection. Here we are given the function F of X is equal to four race to the power of X plus one. And we are also given the function G of X is equal to four raised to the power of negative X plus one. Here we have four answer choice options. ABC and D, each of which display a graph containing both of the functions as well as a statement of their point of intersection. So to begin solving this problem, we first want to begin by considering the first given function which is again, F of X is equal to four raised to the power of X plus one. Well, we need to obtain coordinate points for this function. So we need to consider various arbitrary values of X. So we can first consider when X is equal to negative two. So we need to find F of negative two which gives us four raised to the power of negative two plus one. Now simplifying we get 1/16 plus one which combining these two terms. We get 17 16th, which gives us a decimal equivalent of 1.625. And so now using this X value and the output value, we obtain our first coordinate point which is negative two and 1.625. So now we need to repeat this process and create a table of values for X and Y. So we can choose any values of X, we had negative two, Then we can also choose negative 101 and two. So from inputting negative two into our function F of X, we get again 1.62 five. When we have X is equal to negative one, we have 1.25 from X is equal to zero, we have two, then from X is equal to one, we have five. And lastly from X is equal to two, we get 17. So now we have our table of values. So the next step is to just plot each of these points on a graph. And so plotting, we see we have our five individual points on a graph. And so now with these points, we have begun graphing our function F of X. And so the next step is to draw a smooth curved line through all of these points. And we see we now have our function F of X on a graph here. So now that we have graphed our first function, we can now consider our other function which was G FX is equal to or raise to the power of negative X plus one. So now we are going to repeat the same process for G FX by considering arbitrary values for X. So again, we can consider when X is equal to negative two. So finding G of negative two, we get four raised to the power of negative one times the quantity of negative two plus one. Simplifying we get 16 plus one, which simplifies further to 17. So we see that one of our points is at negative two and 17. So now again, we repeat this procedure and create a table of values. And so we can choose the X values negative two negative 101 and two. And from these X values, we get the Y values of 521.25 and 1.625. So now with this table of values, we can begin graphing our function G FX by first plotting all of these points on a graph. And so here we have a graph containing our five additional points from our G FX function. And we also see that it is on the, the same graph as F of X. So we see that this blue smooth curve line is our function F of X. So now that we have our points from G FX, we just need to draw a smooth curved line that connects all of these points. And in doing so we see we now have our graph of G FX in green as well as our graph of F of X still included in blue. So now with both of our functions graphed, we see that they have a point of intersection at the 0.0.0 and two. So we see that again, we have finished grabbing both of our functions and identified the point of intersection. So with all of this information, we see we are left with answer choice C where again, we have a graph displaying both of our functions and their point of intersection is at zero and two. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Graph y= 2^x and x = 2^y in the same rectangular coordinate system.

Key Concepts

Exponential Functions

Inverse Functions

Graphing in Rectangular Coordinate System

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