Question

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

A. The domain is all real numbers.
B. The range is y > 3.
C. The initial value is 3.
D. The initial value is 9.
E. The simplified base is √2.
F. The simplified base is 3√2.

Answer 1

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.

Step-by-step 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 = 2*9
9 = 3*3

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

Answer 2

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)) = 3*sqrt(1/18) = 3*0.2357 = 0.7071 < 3.

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

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