To determine the equation of the line in function notation, we need to first identify the slope and y-intercept of the line from the graph.

1. Y-Intercept:

Looking at the graph, the line crosses the y-axis at $y = -1$. Therefore, the y-intercept is $-1$.

2. Slope:

To find the slope, we choose two points on the line. The graph clearly shows points at $(0, -1)$ and $(-2, 0)$.

The formula for the slope $m$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using the points $(0, -1)$ and $(-2, 0)$: $m = \frac{0 - (-1)}{-2 - 0} = \frac{1}{-2} = -\frac{1}{2}$

3. Equation of the Line:

The general form of the equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Substituting the slope $-\frac{1}{2}$ and y-intercept $-1$, the equation becomes: $y = -\frac{1}{2}x - 1$ Thus, the correct equation in function notation is: $f(x) = -\frac{1}{2}x - 1$

The correct option is: $\boxed{f(x) = -\frac{1}{2}x - 1}$

Would you like any more details on this process?

Here are 5 related questions to consider:

1. How do you calculate the slope of a line given two points?
2. What is the significance of the y-intercept in a linear equation?
3. How can you determine the equation of a line parallel to a given line?
4. What is the effect of changing the slope in a linear equation?
5. How would the equation change if the line was shifted vertically by 3 units?

Tip: Always ensure you pick points where the line crosses grid points exactly to avoid calculation errors when finding the slope.