To determine which of the given functions grows at the same rate as e^x as x goes to infinity, we need to compare the growth rates of the functions.
The function e^x grows exponentially as x increases. As x goes to infinity, e^x increases without bound.
Let's examine each option one by one:
a) e^(x+3): The exponent x+3 does not affect the growth rate of e^x. As x goes to infinity, this function still grows exponentially and at the same rate as e^x.
b) e^(3x): In this case, the exponent 3x increases at a faster rate compared to just x. As x goes to infinity, e^(3x) grows even faster than e^x. Therefore, e^(3x) does not grow at the same rate as e^x as x goes to infinity.
c) e^(2x): Similar to the previous case, the exponent 2x increases at a faster rate compared to x. As x goes to infinity, e^(2x) grows faster than e^x. Hence, e^(2x) does not grow at the same rate as e^x as x goes to infinity.
d) e^(-x): In this case, the exponent -x decreases as x goes to infinity. As x approaches infinity, e^(-x) tends to approach zero. Therefore, e^(-x) does not grow at the same rate as e^x as x goes to infinity.
Therefore, the answer is option a) e^(x+3), which grows at the same rate as e^x as x goes to infinity.