So in part a of this question we're dealing with why equals two to the X. Power. And so Y equals two to the X. Power is an exponential growth function. Right? Its growth since my base is greater than one, It's passing through the .001 as well as the 0.1 comma two and then two comma four, et cetera. And we want to sketch are inverse. So let's think about that to do that. We're going to switch our role of X. And Y. For each of the points on the graph of y equals two to the X. So 01. That's going to become the 10.1 comma zero one, comma 2. That's going to become the point two comma one and two comma four. That's going to become the 40.4 comma two. And so looking at our inverse of why equals two to the X. Power. We're going to have a graph that looks very much like this right now. If I want to know what my inverse is. Well, the inverse of an exponential function is a log rhythmic function having the same base. And so my inverse in this case which is hearing Green would just be Y equals the log based chew of X. Now on that inverse function, let's talk about several characteristics starting with the domain and range. Well, the domain of this log rhythmic function. My set of X values that I can plug in Is from 0 to Infinity. How about my range? Well my set of why values that can come out of a log arrhythmic function is all reels negative infinity to positive infinity, I believe this log arrhythmic function does have an X intercept that X intercept is right here at the 0.10. Right? So my X intercept is ones you in terms of a Y intercept. I believe there are none. I don't believe there are any Why intercepts here, right? Because this log rhythmic function has a vertical ascent toe along the Y axis. We in fact do have a vertical ascent toe and that vertical ascent tote this time is along the y axis. It is X equals right. And so that was part A. Now in part B we want to do the same thing, but for a different exponential function. So my exponential function now will be y equals 1/3 raised to the X power. This is going to be an exponential decay since my base is now less than one. Now it's still going to pass. Looking at the graph, they gave us through the .01. When my ex is negative one, I get a Y value of three. So this is the point negative 13. And when my ex is positive one, I have a Y value of one third. This is the 10.1 comma one third. And here is this exponential decay function. Okay, now I want to know what would my inverse function look like. Right? And we're doing this on the same set of axes. So we're just going to switch our role of X and Y. So instead of having the .01, I'm going to have the point one comma zero. Instead of having negative one comma three, I'm going to have three common negative one. So I'm gonna have three comma negative one, roughly speaking, that will be somewhere around here and instead of one comma third, I'm going to have a third comma one that's going to be here. And so connecting with a nice smooth curve. My inverse function in blue is going to look something like this. Okay, now here is the inverse. But what is the equation of that in verse? Well, again, the inverse of an exponential function is a log rhythmic function of the same base. And so my inverse function this time would be a log rhythmic function of the same base. It would be Y equals The Log Base 1 3rd of x. Yes, there's my inverse. Now, once I have that in verse again, I want to know some characteristics of this inverse. Starting with the domain and range. Well, the domain of a log rhythmic function. 02 infinity. Right? I can only take the log of something positive. Yeah, my range just like the log arrhythmic function we saw before, negative infinity. To positive infinity. And there's my range in terms of an X intercept, I still have an X intercept? At 10. Right? That was a point on the graph of that log rhythmic function. Why intercepts there were none, Right? Because again, I'm going to have a vertical ascent toe there along the Y axis. And so we do have a vertical ascent over here, just like the other log arrhythmic function. And that vertical ascent toe is at X equals zero. So hopefully, that made sense, and you have a great rest of the day.