Here are the solutions to the problems shown in the image:

58. A car's average gas mileage $G$ as a function of speed $v$:

• Reasonable domain for $f(v)$:
  • The speed $v$ should be positive and within practical limits for a vehicle. Typically, $0 < v \leq 120$ mph is a reasonable assumption, depending on context.

59. Restaurant open from 2 PM to 2 AM with maximum 200 clients:

(a) Reasonable domain for $f(t)$:

• The restaurant operates from 2 PM (t = 0) to 2 AM (t = 12). Thus, $0 \leq t \leq 12$.

(b) Reasonable range for $f(t)$:

• The number of clients can range from 0 to 200. Thus, $0 \leq f(t) \leq 200$.

60. Value $V(a) = 18,000 - 3000a$ for a car $a$ years old:

(a) Domain:

• $a \geq 0$, since the car age cannot be negative. The upper limit depends on when the car value becomes zero:
  $18,000 - 3000a \geq 0$
  $a \leq 6$.
  So, $0 \leq a \leq 6$.

(b) Range:

• When $a = 0$, $V(0) = 18,000$.
  When $a = 6$, $V(6) = 0$.
  Therefore, $0 \leq V(a) \leq 18,000$.

61. Cost $C = 2000 + 4x$ up to $10,000:

(a) Domain:

• The production cost reaches the maximum limit of $10,000:
  $2000 + 4x \leq 10,000$
  $4x \leq 8000$
  $x \leq 2000$.
  Thus, $0 \leq x \leq 2000$.

(b) Range:

• When $x = 0$, $C(0) = 2000$.
  When $x = 2000$, $C(2000) = 10,000$.
  Therefore, $2000 \leq C \leq 10,000$.

Do you need further clarification or detailed steps? Here are 5 related questions to expand the topic:

1. How can domains and ranges change when additional constraints are added to a function?
2. What factors could influence the realistic domain of gas mileage as a function of speed?
3. Could a piecewise function model the restaurant problem more accurately?
4. What happens to the value of a car after 6 years if depreciation isn't linear?
5. How could we model costs that increase quadratically instead of linearly?

Tip: Always visualize the domain and range on a graph to better understand their practical implications!