Let's break down and solve the questions based on the provided image.

Question 1:

Which of the following is the set of all values that the variable $x$ can take?

• a. Range
• b. Intercept
• c. Domain
• d. Zeroes

Answer: The set of all values that the variable $x$ can take is called the domain. Hence, the correct answer is c. Domain.

Question 2:

What is the domain of $f(x) = \frac{x - 3}{x + 3}$?

• a. All real numbers
• b. All real numbers except -3
• c. All real numbers except 3
• d. Cannot be determined

Solution: The domain of a function includes all possible values of $x$ for which the function is defined. Here, the function $f(x) = \frac{x - 3}{x + 3}$ has a denominator $x + 3$. For the function to be defined, the denominator cannot be zero.

Thus, $x + 3 \neq 0 \implies x \neq -3$.

Answer: The domain is all real numbers except $-3$, so the correct answer is b. All real numbers except -3.

Question 3:

What value(s) of $x$ will make the function meaningless for $f(x) = \frac{x - 1}{x}$?

• a. $x = -1$
• b. $x = 0$
• c. $x = 1$
• d. All real numbers

Solution: The function becomes meaningless when the denominator is zero because division by zero is undefined. In $f(x) = \frac{x - 1}{x}$, the denominator is $x$, so we set:

$x \neq 0$

Answer: The value of $x$ that makes the function meaningless is $x = 0$, so the correct answer is b. $x = 0$.

Do you need further details on any of these questions?

Additional Related Questions:

1. What is the difference between domain and range?
2. How do you find the domain of a function with a square root?
3. How does the domain change if there is both a square root and a fraction?
4. What are zeroes of a function and how are they found?
5. Why can't we divide by zero in mathematics?

Tip: To find the domain of a rational function, always check for values of $x$ that make the denominator zero—these values are excluded from the domain.