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Which statements accurately describe the function f(x) = 3√18^x? Check all that apply.

AThe domain is all real numbers.
BThe range is y > 3.
CThe initial value is 3.
DThe initial value is 9.
EThe simplified base is √2.

F. The simplified base is 3√2.

Verified Answer :

nobillionaireNobley

Option A is correct: the domain is all real numbers for there is no real value of x for which f(x) does not have a value.

Option B is incorrect: the range is not y > 3, because, for negative values of x, f(x) < 3, e.g. for x = -1, f(x) = 3 sqrt(18^(-1)) = 3sqrt(1/18) = 30.2357 = 0.7071 < 3.

Option C is correct: f(0) = 3sqrt(18^0) = 3sqrt(1) = 3 1 = 3

Option D is incorrest: (see working for option C)

The correct options are:

A) The domain is all real numbers;
C) The initial value is 3; and 
F) The simplified base is 3√2.

Explanation:

A) The domain is the set of numbers that can be use for x.  In this function, there are no domain restrictions; this means there are no values for x that will not give you a value for y.  This means that all numbers work, which we say is "the set of all real numbers."

C) To find the initial value (starting point), substitute 0 for x:
f(x) = 3(√18)ˣ
f(0) = 3(√18)⁰
f(0) = 3(1) = 3

Thus the initial value is 3.

F)  The base of the current function is √18; this is because that is what is raised to a power.  If we simplify this, we find the prime factorization of 18, looking for pairs of factors to take out.

The prime factorization of 18:
18 = 29
9 = 33

This means 18 = 233; it gives us √18 = √(233).  We have a pair of 3's under the radical, so we take a 3 out; this leaves a 2 under the radical:
√18 = √(233) = 3√2.

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