Answer:
y = 5^x + 2
Explanation:
The function with a range of (2, ∞) is y = 5^x + 2.
To see why, let's analyze each function in turn:
y = 2 ^ x
As x increases, 2^x increases, but it never reaches 2. The range of this function is therefore (0, 2).
y = 2(5 ^ x)
As x increases, 5^x increases and 2(5^x) increases as well. Since 5^x can be arbitrarily large, 2(5^x) can also be arbitrarily large. Therefore, the range of this function is not (2, ∞).
y = 5 ^ (x + 2)
As x increases, 5^(x+2) increases, but it never reaches 5^2 = 25. The range of this function is therefore (0, 25).
y = 5 ^ x + 2
This function can be interpreted in two ways: either as 5^(x+2), or as (5^x) + 2. If we interpret it as 5^(x+2), the range is (5^2, ∞) = (25, ∞). If we interpret it as (5^x) + 2, then the range is (2, ∞).
Therefore, the function with a range of (2, ∞) is y = 5^x + 2.