To find the equation that describes the function using function notation, let's look at the pairs of \(x\) and \(y\) provided:
\[ \begin{align*} x=0 & \quad y=-5 \ x=1 & \quad y=-9 \ x=2 & \quad y=-13 \ x=3 & \quad y=-17 \ \end{align*} \]
The changes in \(y\) as \(x\) increases can help us find the slope of the function.
Calculating the differences in \(y\):
- From \(x=0\) to \(x=1\): \(y\) goes from \(-5\) to \(-9\), a change of \(-4\).
- From \(x=1\) to \(x=2\): \(y\) goes from \(-9\) to \(-13\), a change of \(-4\).
- From \(x=2\) to \(x=3\): \(y\) goes from \(-13\) to \(-17\), a change of \(-4\).
The slope \(m\) is \(-4\).
Now, we can use the slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
Substituting \(m = -4\):
\[ y = -4x + b \]
To find \(b\), we can use one of the points in the table. Using the point \((0, -5)\):
\[ -5 = -4(0) + b \implies b = -5 \]
Now, the equation is:
\[ y = -4x - 5 \]
In function notation, it is expressed as:
\[ f(x) = -4x - 5 \]
Thus, the correct answer is:
f(x) = -4x - 5
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