Question

This table shows some values of the function f(x)=
1
9
2
3
x.
x
–
2
–
1 0 1 2
f(x)
1
4
1
6
1
9
2
27
4
81
Complete the sentences.
When the x-value changes by 1 unit, the function's value changes by
. So, f(x) demonstrates that
functions change by equal
over equal intervals.

Answer 1

this is all wrong

Answer 2

the answer choices are exponential or linear and u have to choose numbers for the first vbox

Answer 3

To complete the sentences, we need to analyze how the function \( f(x) \) changes as \( x \) changes by 1 unit.

From the provided values:

- \( f(-2) = 1 \)
- \( f(-1) = 4 \)
- \( f(0) = 1 \)
- \( f(1) = 6 \)
- \( f(2) = 1 \)
- \( f(3) = 27 \)
- \( f(4) = 81 \)

Let's note the changes in \( f(x) \) for each unit increase in \( x \):

- From \( x = -2 \) to \( x = -1 \): \( f(-1) - f(-2) = 4 - 1 = 3 \)
- From \( x = -1 \) to \( x = 0 \): \( f(0) - f(-1) = 1 - 4 = -3 \)
- From \( x = 0 \) to \( x = 1 \): \( f(1) - f(0) = 6 - 1 = 5 \)
- From \( x = 1 \) to \( x = 2 \): \( f(2) - f(1) = 1 - 6 = -5 \)
- From \( x = 2 \) to \( x = 3 \): \( f(3) - f(2) = 27 - 1 = 26 \)
- From \( x = 3 \) to \( x = 4 \): \( f(4) - f(3) = 81 - 27 = 54 \)

We can see that the change in \( f(x) \) is not consistent but produces varying results:

The overall response can be summarized as follows:

When the x-value changes by 1 unit, the function's value changes by varying amounts, such as 3, -3, 5, -5, 26, and 54 depending on the interval. So, \( f(x) \) demonstrates that **non-linear** functions change by **unequal amounts** over equal intervals.

So the completed sentences would be:

"When the x-value changes by 1 unit, the function's value changes by varying amounts. So, f(x) demonstrates that non-linear functions change by unequal amounts over equal intervals."

Answer 4

I apologize for the confusion in my earlier response. Let's clarify this and take a different approach to analyze the function provided and complete the sentences accordingly.

Here's the table again for reference:

| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|----|----|----|---|---|---|---|---|
| f(x) | 1 | 4 | 1 | 6 | 2 | 27 | 81 |

Given the function \( f(x) \):

1. Find the changes in \( f(x) \) when \( x \) increases by 1 unit:

- From \( x = -2 \) to \( x = -1 \): \( f(-1) - f(-2) = 4 - 1 = 3 \)
- From \( x = -1 \) to \( x = 0 \): \( f(0) - f(-1) = 1 - 4 = -3 \)
- From \( x = 0 \) to \( x = 1 \): \( f(1) - f(0) = 6 - 1 = 5 \)
- From \( x = 1 \) to \( x = 2 \): \( f(2) - f(1) = 2 - 6 = -4 \)
- From \( x = 2 \) to \( x = 3 \): \( f(3) - f(2) = 27 - 2 = 25 \)
- From \( x = 3 \) to \( x = 4 \): \( f(4) - f(3) = 81 - 27 = 54 \)

As seen, the changes vary.

Now we can conclude:

**Completed Sentences:**
"When the x-value changes by 1 unit, the function's value changes by varying amounts, including positive and negative values. So, f(x) demonstrates that non-linear functions change by unequal amounts over equal intervals."

If you have specific values or patterns you would like to highlight in these sentences, please feel free to provide more context or numbers to clarify further. Thank you for your patience!

Answer 5

Thank you for your patience and for providing more context regarding the answer choices. Given the information you have shared, let's analyze the function \( f(x) \) again to determine whether it is linear or exponential.

Looking at the values of \( f(x) \):

- \( f(-2) = 1 \)
- \( f(-1) = 4 \)
- \( f(0) = 1 \)
- \( f(1) = 6 \)
- \( f(2) = 2 \)
- \( f(3) = 27 \)
- \( f(4) = 81 \)

To identify a pattern, we can observe the differences between consecutive values:

- From \( f(-2) \) to \( f(-1) \): \( 4 - 1 = 3 \)
- From \( f(-1) \) to \( f(0) \): \( 1 - 4 = -3 \)
- From \( f(0) \) to \( f(1) \): \( 6 - 1 = 5 \)
- From \( f(1) \) to \( f(2) \): \( 2 - 6 = -4 \)
- From \( f(2) \) to \( f(3) \): \( 27 - 2 = 25 \)
- From \( f(3) \) to \( f(4) \): \( 81 - 27 = 54 \)

The output values change inconsistently, which suggests a non-linear relationship. However, some values seem to suggest an exponential pattern.

To analyze this, we need to focus on the question about how the function changes as \( x \) increases by 1.

### Completing the Sentences

- **When the x-value changes by 1 unit, the function's value changes by a variable amount.** (Here you can put specific numbers, like \( 3, -3, 5, -4, 25, 54 \), based on previous calculations.)
- **So, \( f(x) \) demonstrates that** **exponential** **functions change by equal ratios over equal intervals.**

Thus, it should look like this:

1. **First blank:** Pick one of the specific changes, such as **3, -3, 5, -4, 25, 54** (you can use one of these at your discretion).
2. **Second blank:** Choose **exponential** since the values increase rapidly.

So, the completed sentences could be:

"When the x-value changes by 1 unit, the function's value changes by **3** (or any other number from the changes observed). So, f(x) demonstrates that **exponential** functions change by equal **ratios** over equal intervals."

If you need further adjustments or clarifications, please let me know!