Final answer:
The function f(x) = 2^(-x) + 3 is the result of reflecting the parent exponential function g(x) = 2^x over the y-axis and shifting it up 3 units. It has a domain of all real numbers, a range of y > 3, and a horizontal asymptote at y = 3 with no vertical asymptote.
Step-by-step explanation:
To describe the transformation of the function f(x) = 2^(-x) + 3, we need to understand the effects of each component of this function compared to its parent function. The parent function in this case is g(x) = 2^x, which is an exponential function.
The transformation occurs in two steps:
The negative sign in the exponent '-x' reflects the graph across the y-axis. This means if the graph originally rose from left to right, post-transformation it will decrease from left to right.
The addition of '+3' shifts the graph upwards by three units along the y-axis.
The domain of f(x) is all real numbers because you can input any value of x into the function. The range of f(x) is y > 3, since the graph is shifted up by 3 units and the exponential function approaches but never reaches 0.
The horizontal asymptote of f(x) is y = 3 because as x approaches infinity, the f(x) approaches 3. There is no vertical asymptote for this function because exponential functions do not possess vertical asymptotes; they are defined for all real numbers.