Difficulty: Medium

Average Score: 70%

What is the domain of the function f(x) = (x - 2)/(x² - 9)?

Rationale

All real numbers except 3 and -3.

The function f(x) = (x - 2)/(x² - 9) has a domain that excludes values where the denominator equals zero, which occurs at x = 3 and x = -3. These values create vertical asymptotes, meaning the function is undefined at these points.

A (All real numbers) YOUR ANSWER

This choice is incorrect because the function is not defined for every real number. Specifically, it fails at x = 3 and x = -3, where the denominator becomes zero, leading to undefined behavior. Therefore, the domain cannot include all real numbers.

B (All real numbers except 2 and -3) YOUR ANSWER

This choice is also incorrect because while x = 2 is a value where the function is defined (f(2) = 0), x = -3 is a point where the function is undefined. Consequently, this option misidentifies the critical points that limit the domain.

C (All real numbers except 3 and -3) CORRECT

This is the correct choice. The function f(x) is undefined at x = 3 and x = -3 due to the denominator becoming zero at these values. Thus, the domain must exclude these two points while including all other real numbers.

D (All real numbers except 2, 3, and -3) YOUR ANSWER

This option is incorrect because it unnecessarily excludes x = 2, where the function is defined and equals zero. The correct domain should only exclude the points where the function is undefined, specifically x = 3 and x = -3.

Conclusion

The domain of the function f(x) = (x - 2)/(x² - 9) is determined by the values that make the denominator zero, which are x = 3 and x = -3. Excluding these points, the domain consists of all other real numbers. Understanding this helps in analyzing the behavior of rational functions and their asymptotic properties.

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