Let's break down the problem step by step.

The function given is: $f(x) = 10x + 20$ where $x$ is the number of members on the team and $f(x)$ is the total cost in dollars for buying T-shirts for the team.

Conditions:

• There are at least 9 members but no more than 17 members on the team. Hence, $x$ is constrained by: $9 \leq x \leq 17$

Step 1: Finding the possible range for $f(x)$

For the given values of $x$ (from 9 to 17), we calculate the corresponding values of $f(x)$:

• When $x = 9$: $f(9) = 10(9) + 20 = 90 + 20 = 110$

• When $x = 17$: $f(17) = 10(17) + 20 = 170 + 20 = 190$

Thus, the range of $f(x)$ is: $110 \leq f(x) \leq 190$

Step 2: Analyzing the answer choices

1. "The values of $x$ and $f(x)$ must both be whole numbers between 10 and 20."

   • This is incorrect because the range of $x$ is between 9 and 17, and the range of $f(x)$ is between 110 and 190, not between 10 and 20.

2. "The value of $x$ must be a whole number between 9 and 17 and the value of $f(x)$ must be a whole number between 110 and 190."

   • This is correct. Based on the calculations, $x$ is constrained between 9 and 17, and $f(x)$ is constrained between 110 and 190.

3. "The value of $x$ must be a whole number between 0 and 190 and the value of $f(x)$ must be a whole number between 0 and 17."

   • This is incorrect because the range of $x$ is 9 to 17, not 0 to 190. Also, the range of $f(x)$ is 110 to 190, not 0 to 17.

4. "The value of $x$ must be a whole number between 0 and 17 and the value of $f(x)$ must be a whole number between 0 and 190."

   • This is incorrect because although the range of $x$ is between 9 and 17, the range of $f(x)$ starts from 110, not 0.

Correct answer:

The second statement is the correct one:

• "The value of $x$ must be a whole number between 9 and 17 and the value of $f(x)$ must be a whole number between 110 and 190."

Would you like more details on this solution, or have any questions? Here are some related questions to explore further:

1. How would the function change if the cost per T-shirt was $15 instead of $10?
2. What happens to the range of $f(x)$ if the number of team members increases to a maximum of 20?
3. If the fixed cost (the constant term in the function) were increased, how would that affect $f(x)$?
4. Could the function $f(x)$ represent a scenario where the cost per T-shirt decreases as the number of team members increases?
5. What would be the result of evaluating $f(x)$ for non-integer values of $x$?

Tip: Always check the boundaries of your variables to ensure the solution meets all the constraints given in the problem.