To solve this problem, let's analyze the table and determine the correct equation in function notation.

Step 1: Determine the Slope

The values of $y$ decrease as $x$ increases, suggesting a linear relationship with a negative slope. We can calculate the slope ($m$) between points by using the formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(0, -5)$ and $(1, -9)$: $m = \frac{-9 - (-5)}{1 - 0} = \frac{-4}{1} = -4$

So, the slope of the function is $-4$.

Step 2: Find the Y-Intercept

When $x = 0$, $y = -5$. This value is the y-intercept $b$ of the linear function. Thus, the equation takes the form: $y = -4x - 5$

Step 3: Identify the Correct Option

In function notation, this equation is written as: $f(x) = -4x - 5$

Answer:

The correct option is: $\text{f(x) = -4x - 5}$

This corresponds to the fourth option listed.

Would you like further details on this process, or do you have any other questions?

Here are some related questions for further practice:

1. How would you find the equation if the function had a positive slope?
2. What would the equation look like if the y-intercept were a positive value?
3. How can you verify that a function is linear just by looking at the table values?
4. What other methods can you use to determine the slope from a table?
5. How do you interpret function notation like $f(x)$ in practical terms?

Tip: For any linear function, the slope can be calculated between any two points on the line since the rate of change remains constant.