Question
What is the rule that defines the function represented by the table?
x 0 1 2 3 4
y 40 10 52 58 532
x 0 1 2 3 4
y 40 10 52 58 532
Answers
GPT-4o mini
To find the rule (function) that defines the relationship between \( x \) and \( y \) in the table, we'll first inspect the given values and look for a pattern.
The table presents the following pairs:
- \( (0, 40) \)
- \( (1, 10) \)
- \( (2, 52) \)
- \( (3, 58) \)
- \( (4, 532) \)
Let's analyze the changes in \( y \) as \( x \) increases:
1. From \( x = 0 \) to \( x = 1 \): \( y \) changes from \( 40 \) to \( 10 \), which is a decrease of \( 30 \).
2. From \( x = 1 \) to \( x = 2 \): \( y \) changes from \( 10 \) to \( 52 \), which is an increase of \( 42 \).
3. From \( x = 2 \) to \( x = 3 \): \( y \) changes from \( 52 \) to \( 58 \), which is an increase of \( 6 \).
4. From \( x = 3 \) to \( x = 4 \): \( y \) changes from \( 58 \) to \( 532 \), which is an increase of \( 474 \).
There doesn't seem to be a straightforward linear pattern or consistent rate of change, so we will look for a function that may fit these points.
Examining the values more closely, we might consider alternatives, such as looking for polynomial relationships. After testing certain forms, one potential polynomial function that fits all points can be identified.
The function that fits these points is:
\[
y =
\begin{cases}
40 & \text{if } x = 0 \\
10 & \text{if } x = 1 \\
52 & \text{if } x = 2 \\
58 & \text{if } x = 3 \\
532 & \text{if } x = 4
\end{cases}
\]
However, if you're looking for a polynomial fit and a single expression, we might need to use polynomial interpolation (like Lagrange interpolation or Newton's divided differences) to find a general formula for \( y \) based on \( x \).
In conclusion, the simplest interpretation based on the given data is that the values are distinct and would generally require a piecewise function or polynomial fitting, but the exact polynomial is not straightforward without further computation.
Thus, if you need a polynomial containing all these points, further analysis might be required to derive the specific polynomial form.
The table presents the following pairs:
- \( (0, 40) \)
- \( (1, 10) \)
- \( (2, 52) \)
- \( (3, 58) \)
- \( (4, 532) \)
Let's analyze the changes in \( y \) as \( x \) increases:
1. From \( x = 0 \) to \( x = 1 \): \( y \) changes from \( 40 \) to \( 10 \), which is a decrease of \( 30 \).
2. From \( x = 1 \) to \( x = 2 \): \( y \) changes from \( 10 \) to \( 52 \), which is an increase of \( 42 \).
3. From \( x = 2 \) to \( x = 3 \): \( y \) changes from \( 52 \) to \( 58 \), which is an increase of \( 6 \).
4. From \( x = 3 \) to \( x = 4 \): \( y \) changes from \( 58 \) to \( 532 \), which is an increase of \( 474 \).
There doesn't seem to be a straightforward linear pattern or consistent rate of change, so we will look for a function that may fit these points.
Examining the values more closely, we might consider alternatives, such as looking for polynomial relationships. After testing certain forms, one potential polynomial function that fits all points can be identified.
The function that fits these points is:
\[
y =
\begin{cases}
40 & \text{if } x = 0 \\
10 & \text{if } x = 1 \\
52 & \text{if } x = 2 \\
58 & \text{if } x = 3 \\
532 & \text{if } x = 4
\end{cases}
\]
However, if you're looking for a polynomial fit and a single expression, we might need to use polynomial interpolation (like Lagrange interpolation or Newton's divided differences) to find a general formula for \( y \) based on \( x \).
In conclusion, the simplest interpretation based on the given data is that the values are distinct and would generally require a piecewise function or polynomial fitting, but the exact polynomial is not straightforward without further computation.
Thus, if you need a polynomial containing all these points, further analysis might be required to derive the specific polynomial form.