Hey, everyone. In this problem, we're going to work through graphing the function f of x is equal to \( \frac{1}{2} \) to the power of x together. So let's go ahead and get started. Now we're just going to go through and plot some points by plugging in some values to our function. So let's start with x equals 0 and plug 0 into our function and see what we get. So if I take \( \frac{1}{2} \) and raise it to the power of 0, I know that anything raised to the power of 0 is 1 regardless of if it's a fraction or not. So my very first point is at 1, and I can go ahead and plot that on my graph. Then if I plug 1 in for x, I get \( \frac{1}{2} \) to the power of 1, which if it's raised to the power of 1, it just gets itself. So this is simply \( \frac{1}{2} \), and I could plot my second point at 1, \( \frac{1}{2} \). Then for 2, x equals 2, plugging that into my function, I get \( \frac{1}{2} \) raised to the power of 2. Now one raised to the power of 2 is just 1, so really I'm just doing \( \frac{1}{2^2} \), which is going to give me \( \frac{1}{4} \). So we're getting much smaller here, to \( \frac{1}{4} \) getting close to that x axis. And then finally, for x equals 3, if I put \( \frac{1}{2} \) and raise that to the power of 3, that's going to give me \( \frac{1}{8} \), which again is getting really small and close to that x axis over here. Now looking at our negative numbers, if I start to plug in some negative numbers for x starting with negative one, if I take \( \frac{1}{2} \) and raise it to the power of negative one, whenever we deal with whole numbers like 2 and raise them to a negative power, we end up with a fraction. Whenever we take a fraction and raise it to a negative power, we are instead going to end up with a whole number. So this is simply going to give me 2 because I just flipped the fraction. I get 2 over 1. That's just 2. So I can plot this point at negative 1, 2. Then if I take \( \frac{1}{2} \) and raise it to the power of negative 2, I'm going to get 4. So I can plot this point at negative 2, 4. Then for my last negative point, if I take \( \frac{1}{2} \) and raise it to the power of negative 3, I end up with 8 and I can, of course, go ahead and plot that point on my graph. Now we see that we have all of these whole numbers as we get into the negatives and then we had their corresponding fractions. We just flipped them under to the bottom of a fraction, and they're the same numbers because we're dealing with the same powers. Some are positive and some are negative. So some of these values will be fractions and some will be whole numbers, just like we saw for 2 to the power of x. Now, if you think about it, \( \frac{1}{2} \) to the power of x is really just 2 to the power of negative x. So that's why we're seeing so many similarities here. Let's go ahead and connect our graph. So on this side, going up and then on my right side here, I see that I'm getting really close to that x axis, not quite touching it though, which tells me I'm dealing with an asymptote. So I can go ahead and plot my asymptote here on this x axis because it is a horizontal asymptote right at y equals 0, and we can go ahead and denote that here our asymptote at y equals 0 plotted, of course, using a dashed line because it's an asymptote.

Now looking at our graph here, you might notice that it looks really similar to the graph of 2 to the power of x, which looks something like this. So now our graph of \( \frac{1}{2} \) to the power of x is kind of just this flipped, which makes perfect sense because we just said that this \( \frac{1}{2} \) to the power of x is the same exact thing as 2 to the power of negative x, so it makes sense that it's the same graph but flipped because it just underwent a transformation due to this negative sign here. Now let's go ahead and finish up here and get the rest of the information for this graph, like the domain and the range. So here, our domain of every single exponential function is always going to be the same. So here, it is the same as before, all real numbers. Now our range is dependent on where our asymptote is. And since our asymptote is again at y equals 0 and our graph is completely above that, we know that our range is going to go from our asymptote at 0 until infinity. So our range is 0 to infinity, using parentheses because the 0 is not included. Now that we have the entire picture of our graph of a \( \frac{1}{2} \) to the power of x, let's get some more practice.