Question 30

The domain of the function f(x) = (x - 2)/(x² - 9) is all real numbers except 3 and -3.

The function f(x) is defined for all real numbers except where the denominator equals zero. In this case, the denominator x² - 9 equals zero when x = 3 and x = -3, meaning these values are excluded from the domain.

A) All real numbers

This option is incorrect because it suggests that the function is defined for all real numbers. However, the function is undefined at x = 3 and x = -3 due to the denominator becoming zero at these points.

B) All real numbers except 2 and -3

This option is incorrect as it mistakenly excludes 2, which does not cause the denominator to be zero. The only values that need to be excluded from the domain are 3 and -3.

C) All real numbers except 3 and -3

This option is correct because the function is undefined at x = 3 and x = -3, which are the points where the denominator becomes zero. Thus, these are the only values excluded from the domain.

D) All real numbers except 2, 3, and -3

This option is incorrect as it incorrectly excludes 2. The function is defined at x = 2, while it is only undefined at x = 3 and x = -3.

Conclusion

The correct answer is C, as the function is defined for all real numbers except at the points where the denominator becomes zero, specifically x = 3 and x = -3. All other options fail to accurately identify the values that must be excluded from the domain.