Final answer:
The domain of the function f(x) = log4(x+2) + 3 is all real numbers greater than -2, and the range is all real numbers.
Step-by-step explanation:
To find the domain and range of the function f(x) = log4(x+2) + 3, we must consider the properties of logarithmic functions. The argument of the logarithm function (x+2 in this case) must be positive, which means x+2 > 0 or x > -2. Therefore, the domain of f(x) is all real numbers greater than -2, written in interval notation as (-2, ∞).
The range of logarithmic functions is all real numbers, since the output of a log function can be any real number. Adding 3 to a logarithm shifts the graph vertically but does not affect the range. Hence, the range of the given function is also all real numbers, which can be represented as (-∞, ∞).
As noted, as x increases, log(x) increases as well. This behavior is consistent across all base logarithms, including log4(x).