Question 18

The range of f(x) = 1/x is all real numbers except 0.

The function f(x) = 1/x is defined for all real numbers except x = 0, and it takes on all real values except for 0. As x approaches 0 from either side, f(x) diverges to positive or negative infinity, indicating that 0 is never reached in the range.

A) All real numbers

This option is incorrect because the function f(x) = 1/x cannot equal 0 for any value of x. While it approaches 0 as x approaches infinity or negative infinity, it never actually attains that value, thus excluding it from the range.

B) All real numbers except 0

This option is correct as the function takes every real number value except for 0. As x varies over all non-zero real numbers, f(x) will produce every real number value, demonstrating that the only exclusion from the range is 0 itself.

C) All positive numbers

This option is incorrect because f(x) = 1/x produces both positive and negative values for positive and negative x, respectively. Therefore, the range cannot be limited to only positive numbers, as it also includes negative values.

D) All negative numbers

This option is also incorrect since f(x) = 1/x includes positive values when x is positive. Thus, the range cannot be confined to just negative numbers, as it encompasses both positive and negative real numbers, excluding only 0.

Conclusion

The correct answer is that the range of f(x) = 1/x is all real numbers except 0. This is because the function can take any real number value while asymptotically approaching 0 but never reaching it, which is why all other options fail to accurately describe the range.