If f(x) = 1/x, which of the following describes the range of f(x)?

Rationale

All real numbers except 0 describe the range of f(x).

The function f(x) = 1/x is defined for all real numbers except x = 0, and its output can take any real value except zero, as the function approaches but never reaches zero. This characteristic ensures that the range is all real numbers excluding 0.

A (All real numbers) YOUR ANSWER

This option incorrectly includes 0 in the range of the function. Since f(x) approaches 0 as x approaches infinity or negative infinity, but never actually reaches it, 0 cannot be part of the range. Therefore, the range cannot be all real numbers.

B (All real numbers except 0) CORRECT

Correctly identifies the range of f(x). The function takes on every value in the real number line, but since f(x) = 1/x cannot equal 0 for any real x, the range is indeed all real numbers except for 0.

C (All positive numbers) YOUR ANSWER

This choice restricts the range to only positive values, which is not accurate. For negative values of x, f(x) yields negative outputs, demonstrating that the function also produces negative results, thereby invalidating this option.

D (All negative numbers) YOUR ANSWER

This option is also incorrect as it limits the range exclusively to negative values. The function produces positive outputs for positive inputs (x > 0), meaning the range includes both positive and negative numbers, not just negative ones.

Conclusion

The function f(x) = 1/x has a range that encompasses all real numbers except for 0, as it can produce both positive and negative values for appropriate inputs. Understanding the behavior of f(x) reveals that 0 is asymptotically approached but never attained, defining the range as all real numbers excluding this value.

If f(x) = 1/x, which of the following describes the range of f(x)?

All real numbers except 0 describe the range of f(x).

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