Given that $f(x)$ is a linear function, its equation can be written in the form:

$f(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Find the slope ($m$)

The slope of a linear function is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

We are given two points: $(-5, 0)$ and $(2, 1)$. Substituting these values into the slope formula:

$m = \frac{1 - 0}{2 - (-5)} = \frac{1}{7}$

So, the slope $m = \frac{1}{7}$.

Step 2: Find the y-intercept ($b$)

Now that we know the slope, we can use one of the given points to find the y-intercept $b$. Let's use the point $(2, 1)$.

Substitute $m = \frac{1}{7}$, $x = 2$, and $f(x) = 1$ into the equation $f(x) = mx + b$:

$1 = \frac{1}{7}(2) + b$

$1 = \frac{2}{7} + b$

Now, solve for $b$:

$b = 1 - \frac{2}{7} = \frac{7}{7} - \frac{2}{7} = \frac{5}{7}$

Step 3: Write the equation of the function

Now that we have both $m$ and $b$, the equation for $f(x)$ is:

$f(x) = \frac{1}{7}x + \frac{5}{7}$

Would you like further details or clarification?

Here are 5 related questions for further exploration:

1. How would the equation change if one of the points were different?
2. Can you express this function in terms of another point-slope form?
3. What does the graph of this function look like?
4. How do you find the x-intercept of this function?
5. How would you calculate the slope if the function were non-linear?

Tip: For any linear function, the slope represents the rate of change between two points.