Final answer:
The domain being natural numbers and the recursive formula F(x+1) = Three-halves(F(x)) are true. The range being natural numbers is not true since it includes 13.5, which isn't natural. The corrected explicit formula and the exponential growth statement are also true.
Step-by-step explanation:
For the 4 points plotted on the coordinate plane: (1, 4), (2, 6), (3, 9), (4, 13.5), we must identify which statements are true regarding the geometric sequence they represent.
The domain consists of the x-values of the points, which are 1, 2, 3, and 4. These values represent specific inputs into a function, and they are natural numbers, so the statement that the domain is the set of natural numbers is true.
The range is comprised of the y-values of the points, which are 4, 6, 9, and 13.5. Since 13.5 is not a natural number, the statement that the range is the set of natural numbers is false.
To determine if the recursive formula is correct, we must check if applying the formula f(x + 1) = Three-halves(f(x)) results in the sequence of y-values. Starting with f(1) = 4:
- f(2) should equal Three-halves(4) = 6, which matches the given point (2, 6).
- f(3) should equal Three-halves(6) = 9, matching (3, 9).
- f(4) should equal Three-halves(9) = 13.5, matching (4, 13.5).
Therefore, the given recursive formula is true.
For the explicit formula f(x) = 4(three-halves) Superscript x, we notice that this is not a correctly written exponential formula, and when the correct exponent notation is used, f(x) = 4 * (3/2)^(x-1), it properly calculates the y-values for the given x-values:
- f(1) = 4 * (3/2)^(1-1) = 4 * (3/2)^0 = 4
- f(2) = 4 * (3/2)^(2-1) = 4 * (3/2) = 6
- f(3) = 4 * (3/2)^(3-1) = 4 * (9/4) = 9
- f(4) = 4 * (3/2)^(4-1) = 4 * (27/8) = 13.5
The correct explicit formula is true.
Lastly, since each term in the sequence is getting progressively larger, and each y-value is a result of raising a constant to a power multiplied by the initial term, it's indicative of exponential growth, which makes this statement true.