The function provided in the image is:

$f(r) = \sqrt{r + 7} + 1$

We need to evaluate this function at the given values of $r$:

a. $f(-7)$

Substitute $r = -7$ into the function:

$f(-7) = \sqrt{-7 + 7} + 1 = \sqrt{0} + 1 = 0 + 1 = 1$

Thus, $f(-7) = 1$.

b. $f(74)$

Substitute $r = 74$ into the function:

$f(74) = \sqrt{74 + 7} + 1 = \sqrt{81} + 1 = 9 + 1 = 10$

Thus, $f(74) = 10$.

c. $f(x - 7)$

Substitute $r = x - 7$ into the function:

$f(x - 7) = \sqrt{(x - 7) + 7} + 1 = \sqrt{x} + 1$

So, $f(x - 7) = \sqrt{x} + 1$.

Let me know if you'd like more details or have any questions.

Here are five follow-up questions:

1. How do you simplify square roots of negative numbers?
2. Can the function $f(r) = \sqrt{r + 7} + 1$ ever have a negative value?
3. How would you interpret $f(x - 7)$ in terms of graph transformation?
4. What is the domain of the function $f(r)$?
5. How does the square root function affect the behavior of the function as $r$ increases?

Tip: Always check if the expression inside the square root is non-negative before simplifying.