To determine which equation correctly describes the function in the table, let's analyze the data points:
- For \( x = 0 \), \( y = -2 \)
- For \( x = 2 \), \( y = 4 \)
- For \( x = 4 \), \( y = 10 \)
- For \( x = 6 \), \( y = 16 \)
We can attempt to derive a linear function that fits these points.
Let's first calculate the differences in \( y \) as \( x \) increases:
- From \( x = 0 \) to \( x = 2 \): \( 4 - (-2) = 6 \)
- From \( x = 2 \) to \( x = 4 \): \( 10 - 4 = 6 \)
- From \( x = 4 \) to \( x = 6 \): \( 16 - 10 = 6 \)
The differences in \( y \) are consistent, indicating the function is linear. The slope \( m \) can be determined as follows: \[ m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3 \]
Using the slope and one of the points, we can use point-slope form to find the equation of the line. Using the point \( (0, -2) \):
The slope-intercept form is: \[ y = mx + b \]
Substituting \( m = 3 \): \[ y = 3x + b \]
To find \( b \), we can use the point \( (0, -2) \): \[ -2 = 3(0) + b \] \[ b = -2 \]
Thus, the equation is: \[ y = 3x - 2 \]
Now, in function notation, we can express this as: \[ f(x) = 3x - 2 \]
So, the correct answer is:
- f(x) = 3x - 2 (f of x is equal to 3 x minus 2)
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