Question

Answer the questions below to determine what kind of function is depicted in the table below.
xx 00 11 22 33 44
f, of, xf(x) 88 1616 2424 3232 4040
Answer
Attempt 1 out of 2

This function is
because
.

Answer 1

thats not right
try again

Answer 2

Let’s analyze the table provided:

The values for \( x \) are:
- \( 0, 1, 2, 3, 4 \)

The corresponding values for \( f(x) \) are:
- \( 88, 1616, 2424, 3232, 4040 \)

To see what kind of function this might be, we need to determine if there is a pattern between \( x \) and \( f(x) \).

1. **Look for patterns**:
- First differences:
- \( f(1) - f(0) = 1616 - 88 = 1528 \)
- \( f(2) - f(1) = 2424 - 1616 = 808 \)
- \( f(3) - f(2) = 3232 - 2424 = 808 \)
- \( f(4) - f(3) = 4040 - 3232 = 808 \)

- The first difference changes from \( 1528 \) to \( 808 \) but stabilizes at \( 808 \) from \( x=2 \) onward.

- Second differences:
- \( 808 - 1528 = -720 \)
- \( 808 - 808 = 0 \)
- \( 808 - 808 = 0 \)

2. **Conclusion from differences**:
- Since the first differences stabilize and the second differences are constant (0), this indicates that the function is quadratic.

Thus, we can conclude:

**Answer**:
This function is quadratic because the second differences are constant and the first differences stabilize after the initial value.

Answer 3

I apologize for the confusion. Let's take a closer look at the data you provided to determine the nature of the function accurately.

The input-output pairs you provided are:

\[
\begin{array}{c|c}
x & f(x) \\
\hline
0 & 88 \\
1 & 1616 \\
2 & 2424 \\
3 & 3232 \\
4 & 4040 \\
\end{array}
\]

To analyze the numbers, let's observe how \( f(x) \) relates to \( x \):

1. **Notice the pattern in the outputs**:
- **When \( x = 0 \), \( f(0) = 88 \)**
- **When \( x = 1 \), \( f(1) = 1616 \)**
- **When \( x = 2 \), \( f(2) = 2424 \)**
- **When \( x = 3 \), \( f(3) = 3232 \)**
- **When \( x = 4 \), \( f(4) = 4040 \)**

2. **Explore the pattern further**:
If we examine the outputs, we can see the following relationship:

- \( f(x) = 808x + 88 \)

3. **Verify it**:
- For \( x = 0 \): \( 808(0) + 88 = 88 \)
- For \( x = 1 \): \( 808(1) + 88 = 896 \) [This should be 1616 so let’s retry the analysis]

The values \( f(x) = 808x + 88 \) doesn’t seem to align according to initial assumption – let’s try a different angle.

### Correct Evaluation
Looking back at the values:
- \( 0 \rightarrow 88 = 88 \)
- \( 1 \rightarrow 1616 = 88 \times 18 \)
- \( 2 \rightarrow 2424 = 88 \times 27 \)
- \( 3 \rightarrow 3232 = 88 \times 36 \)
- \( 4 \rightarrow 4040 = 88 \times 45 \)

It becomes more clear that each subsequent value is created by multiplying \( 88 \) by increasing multiples of \( 9 \):
- \( 88 \times (1 \times 9) = 88 \)
- \( 88 \times (2 \times 9) = 1616 \)
- \( 88 \times (3 \times 9) = 2424 \)

Therefore, it seems like we might actually have a linear function based upon constant multiplication.

### Corrections:
Upon reevaluation, we can represent the function as follows:
- The function appears linear in form:

### Final Conclusion:

This function is linear because the output values can be described by a simple formula, \( f(x) = 88 + (x + 1) \times 808 \). The outputs increase by a constant pattern based on multiples.

So, the filling in would be:

**This function is linear because the output increases consistently with each increment in \( x \) by a fixed amount.**

I appreciate your patience! If there is still anything unclear or further corrections needed, please let me know!