Part 1Determine whether the function given in the table is linear, exponential, or neither. If the function is linear, find a linear function that models the data; if it is exponential, find an exponential function that models the data. Part 1[table]
x | f(x) minus−1 | nine eighths98 0 | 99 1 | 7272 2 | 576576 3 | 46084608
[/table] Part 1Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The function is exponential. An exponential function that models the data is f(x)equals=[input]enter your response here .(Simplify your answer.) B. The function is linear. A linear function that models the data is f(x)equals=[input]enter your response here .(Simplify your answer.) C. The function is neither linear nor exponential.

Question

Part 1Determine whether the function given in the table is​ linear, exponential, or neither. If the function is​ linear, find a linear function that models the​ data; if it is​ exponential, find an exponential function that models the data. Part 1[table]
x | ​f(x) minus−1 | nine eighths98 0 | 99 1 | 7272 2 | 576576 3 | 46084608
[/table] Part 1Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. The function is exponential. An exponential function that models the data is ​f(x)equals=[input]enter your response here .​(Simplify your​ answer.) B. The function is linear. A linear function that models the data is ​f(x)equals=[input]enter your response here .​(Simplify your​ answer.) C. The function is neither linear nor exponential.

BlackTom AI · Accepted Answer

We begin by restating the question and the provided data to ensure we’re evaluating the right information.
Question and data: Determine whether the table-based function is linear, exponential, or neither. The table shows pairs (x, f(x)) as follows: x = -1 -> f(x) = 9/8, x = 0 -> f(x) = 9, x = 1 -> f(x) = 72, x = 2 -> f(x) = 576, x = 3 -> f(x) = 4608. The task also asks us to give a model if it’s linear or exponential.
Option A states: The function is exponential. An exponential function that models the data is f(x) = 9·8^x.
Option B states: The function is linear. A linear function that models the data is f(x) = [input].
Option C states: The function is neither linear nor exponential.

Analysis of each option:
- Option A (exponential with f(x) = 9·8^x):
  Let’s test the exponential pattern by looking at successive values. Starting from x = 0 to x = 1, f(1) = 72 and f(0) = 9, which is a factor of 8: 72 ÷ 9 = 8. From x = 1 to x = 2, f(2) = 576 and f(1) = 72, which again is a factor of 8: 576 ÷ 72 = 8. From x = 2 to x = 3, f(3) = 4608 and f(2) = 576, giving 4608 ÷ 576 = 8. Even at x = -1, f(-1) = 9/8, and f(0) = 9, giving 9 ÷ (9/8) = 8. The consistent multiplicative factor of 8 between consecutive x-values is a hallmark of exponential growth with base 8, and the explicit form f(x) = 9·8^x matches all the given values. Therefore, Option A is consistent with an exponential model.
- Option B (linear):
  If the function were linear, differences f(x+1) − f(x) should be constant for all x. Compute a few differences: f(0) − f(-1) = 9 − 9/8 = 63/8 ≈ 7.875. Next, f(1) − f(0) = 72 − 9 = 63. Then f(2) − f(1) = 576 − 72 = 504, and f(3) − f(2) = 4608 − 576 = 4032. The successive differences are not constant; they increase dramatically, which contradicts linear behavior. Thus, the data do not fit a linear model.
- Option C (neither linear nor exponential):
  Since we observed a constant multiplicative factor between successive f(x) values (each step multiplies by 8), the data align with an exponential pattern rather than a neither case. Therefore, claiming it’s neither would be incorrect given the observed ratio consistency.

In summary, Option A’s characterization of the data as exponential with f(x) = 9·8^x aligns with the ratios between consecutive f(x) values and matches all provided points, while Option B fails the linear-difference test, and Option C contradicts the clear exponential pattern observed.

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