To graph the function \( g(x) = -\frac{1}{2^x} + 3 \) as a transformation of the parent function \( f(x) = \frac{1}{2^x} \), we start by understanding the transformations involved. The negative sign indicates a reflection over the x-axis, while the "+3" indicates a vertical shift upwards by 3 units.

First, we identify the parent function \( f(x) = \frac{1}{2^x} \). This function has a horizontal asymptote at \( y = 0 \). To graph it, we plot key points: for \( x = 0 \), \( f(0) = 1 \); for \( x = 1 \), \( f(1) = \frac{1}{2} \); and for \( x = -1 \), \( f(-1) = 2 \). Connecting these points gives us the curve of the parent function, which approaches the horizontal asymptote at \( y = 0 \).

Next, we apply the transformations to obtain \( g(x) \). The first step is to shift the horizontal asymptote from \( y = 0 \) to \( y = 3 \) due to the "+3" in the function. This new asymptote is represented by a dashed line at \( y = 3 \).

Since there is a negative sign outside the function, we reflect the points of the parent function over the x-axis. The reflection of the points we plotted earlier results in new coordinates: \( (0, -1) \), \( (1, -\frac{1}{2}) \), and \( (-1, -2) \). After reflecting, we then shift these points vertically upwards by 3 units. This results in the final points: \( (0, 2) \), \( (1, 2.5) \), and \( (-1, 1) \).

Now, we can sketch the graph of \( g(x) \) by connecting these transformed points with a smooth curve that approaches the new asymptote at \( y = 3 \). The graph will be below this asymptote, indicating that the function values are less than 3.

Finally, we determine the domain and range of the function. The domain of \( g(x) \) remains all real numbers, as exponential functions are defined for all \( x \). The range, however, is limited by the horizontal asymptote. Since the function is below the asymptote at \( y = 3 \), the range is from negative infinity up to, but not including, 3. Thus, the range is expressed as \( (-\infty, 3) \).

In summary, the graph of \( g(x) = -\frac{1}{2^x} + 3 \) is a reflection of the parent function \( f(x) = \frac{1}{2^x} \) that has been shifted upwards by 3 units, with a horizontal asymptote at \( y = 3 \), a domain of all real numbers, and a range of \( (-\infty, 3) \).