In Exercises 25-34, begin by graphing f(x) = 2^x. Then use transf...

Question

In Exercises 25-34, begin by graphing f(x) = 2^x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2^x – 1

Accepted Answer

Hello. Today we're going to be graphing the function G of X as a transformation of F of X. And we're also going to be identifying the domain and range of G of X. So what we are given is F of X is equal to tend to the power of X. Now this is an exponential function in standard form And this is where the exponential function passes through the .1 comma 10. Now here we're going to have a horizontal ascent. Tote of Y is equal to zero. And the shape of the graph is going to be a graph that approaches from negative infinity very close to the ascent tote and exponentially increases through one comma 10. So this is going to be the rough shape of F of X. Now that we know what F of X looks like, let's go ahead and identify the transformations we need in order to go from F of X to G of X. G of X is given to us as G of X is equal to 10 to the power of X -9. And here we're only going to need one transformation. And that revolves around the quantity X -9, C, x minus nine is in the form X minus H. And whenever you have a quantity X minus H in a function that indicates a horizontal shift here, H is going to equal to nine. And since H is equal to nine, we're going to have a horizontal shift of nine units to the right? So with that being said, let's go ahead and apply the transformation to the F of X function. And we're going to use the 0.1 comma 10 as a reference point. See if we're going to move one comma 10, 9 units to the right, we're going to add nine to the X value of one comma 10 and that's going to give us the 100.10 comma 10. So let's go ahead and plot that point. So we have this new point which is out here at 10:10. And because this is a horizontal shift, this is not going to affect the horizontal as in total. So we still have a horizontal as em Todas, Y is equal to zero. Furthermore, this type of transformation doesn't affect the shape of the graph. So the shape of the graph is going to remain the same. So this is going to be the rough shape of G of X. Now that we have the function G of X. Let's go ahead and identify the domain and range. So the domain focuses on the least possible X value to the highest possible X value. So we're going to be reading this graph going from left to right now, as you can see this graph starts at negative infinity and then as it's increasing, it's going to continue along the X axis to positive infinity. So that means any value of X is going to work with this graph. Therefore the domain is going to be from negative infinity to positive infinity. Now let's go ahead and identify the range. The range focuses on the lowest possible Y value, to the highest possible Y value. And here we have a horizontal as a tote of Y is equal to zero. What this means is as this graph is decreasing, it's going to get closer and closer to Y is equal to zero but never actually equal zero. So the range is going to start at zero. But since we're not including that value, we're going to assign a parentheses. And as this graph increases infinitely is going to approach positive infinity along the Y axis. So the range is going to go from zero to positive infinity. So now that we have the domain and range and the shape of G of X. We just need to go ahead and select the option that accurately represents this transformation and that option is going to be a. So with that being said, the answer to this problem is a So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video

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