To find a function rule that fits the given points \((4, -5)\), \((8, -1)\), \((9, 0)\), and \((10, -1)\), we can start by observing the points to see if they might fit a polynomial function, particularly a quadratic function, as we have four points.
A quadratic function can be expressed in the form:
\[ f(x) = ax^2 + bx + c \]
where \(a\), \(b\), and \(c\) are constants.
Given that we have four points, we can set up a system of equations based on these points to solve for \(a\), \(b\), and \(c\).
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For the point \((4, -5)\): \[ 16a + 4b + c = -5 \quad \text{(1)} \]
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For the point \((8, -1)\): \[ 64a + 8b + c = -1 \quad \text{(2)} \]
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For the point \((9, 0)\): \[ 81a + 9b + c = 0 \quad \text{(3)} \]
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For the point \((10, -1)\): \[ 100a + 10b + c = -1 \quad \text{(4)} \]
We can solve this system of equations. We will first use equations (1), (2), and (3) to find \(a\), \(b\), and \(c\).
Subtract equation (1) from equation (2): \[ (64a + 8b + c) - (16a + 4b + c) = -1 + 5 \] \[ 48a + 4b = 4 \quad \Rightarrow \quad 12a + b = 1 \quad \text{(5)} \]
Subtract equation (2) from equation (3): \[ (81a + 9b + c) - (64a + 8b + c) = 0 + 1 \] \[ 17a + b = 1 \quad \text{(6)} \]
Now we have two new equations (5) and (6):
- \(12a + b = 1\)
- \(17a + b = 1\)
Subtract equation (5) from equation (6): \[ (17a + b) - (12a + b) = 1 - 1 \] \[ 5a = 0 \quad \Rightarrow \quad a = 0 \]
Now substitute \(a = 0\) back into equation (5): \[ 12(0) + b = 1 \quad \Rightarrow \quad b = 1 \]
Now substitute \(a = 0\) and \(b = 1\) back into equation (1): \[ 16(0) + 4(1) + c = -5 \quad \Rightarrow \quad 4 + c = -5 \quad \Rightarrow \quad c = -9 \]
Thus, the function rule is: \[ f(x) = 0x^2 + 1x - 9 \quad \Rightarrow \quad f(x) = x - 9 \]
Let's verify if the function \(f(x) = x - 9\) fits all given points:
- For \(x = 4\): \(f(4) = 4 - 9 = -5\) ✔
- For \(x = 8\): \(f(8) = 8 - 9 = -1\) ✔
- For \(x = 9\): \(f(9) = 9 - 9 = 0\) ✔
- For \(x = 10\): \(f(10) = 10 - 9 = 1\) ✔
The point \((10, -1)\) does not fit, suggesting that either the function might not be completely indicative of the pattern (which does also indicate that the fourth point likely follows a different rule or measures from a different context).
The derived function \(f(x) = x - 9\) gives correct outputs for the first three points but fails to encapsulate the last point as initially posited. Therefore, while \(f(x) = x - 9\) represents a viable function rule for a substantial subset of the points, further investigation would be needed to accurately seed the fourth point's function dependency or context.
To finalize, a polynomial attempt results in the conclusion: The simplest function rule compliant with three of four points is: \[ f(x) = x - 9 \] with context-aware adjustments potentially necessary considering outlier behavior in fourth point.
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