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Answer the questions below to determine what kind of function is depicted in the table below.
xx 11 22 33 44 55
f, of, xf(x) minus, 4−4 minus, 5−5 minus, 9−9 minus, 27−27 minus, 123−123
Answer
Attempt 1 out of 2

This function is
because

Answer 1

To determine the type of function depicted in the given table, we need to analyze the relationship between the inputs \( x \) and the outputs \( f(x) \).

Here's the provided data:

- For \( x = 1 \), \( f(x) = -4 \)
- For \( x = 2 \), \( f(x) = -5 \)
- For \( x = 3 \), \( f(x) = -9 \)
- For \( x = 4 \), \( f(x) = -27 \)
- For \( x = 5 \), \( f(x) = -123 \)

Now, let's consider whether this function is linear, quadratic, exponential, etc.

1. **Linear Function:** A linear function has a constant rate of change (i.e., the same difference between consecutive outputs). The differences between the outputs are:
- From -4 to -5: Difference = -1
- From -5 to -9: Difference = -4
- From -9 to -27: Difference = -18
- From -27 to -123: Difference = -96

These differences are not constant, so it is not linear.

2. **Quadratic Function:** A quadratic function has an output that can often be expressed in the form \( ax^2 + bx + c \). To see if the second differences are constant, we calculate first differences and then second differences:
- First differences from above: -1, -4, -18, -96
- Differences of those differences (second differences):
- From -1 to -4: Difference = -3
- From -4 to -18: Difference = -14
- From -18 to -96: Difference = -78

The second differences are also not constant, so it is not quadratic.

3. **Exponential Function:** An exponential function has outputs that grow (or decay) by multiplicative factors. To verify, we would look for a constant ratio.

Calculations show that the outputs are decreasing rapidly, suggesting they may follow a pattern of growth or decay, possibly resembling an exponential decay.

Since we don't specifically identify a defined exponential function in the differences or values, we can say that the function resembles more complex forms. From the values given, a polynomial of degree greater than 2 or other specialized functions might also describe this relationship.

However, with the information here, it's safe to broadly classify the function as **non-linear** due to the non-constant difference patterns, and it may resemble some form of a complex or exponential function.

**Final Answer**: This function is non-linear because the differences between \( f(x) \) values are not constant and do not follow a simple polynomial or linear pattern.