How do you determine the range of f(x) = 1/x?

The range of \( f(x) = \frac{1}{x} \) is all real numbers except 0.

To determine the range of the function \( f(x) = \frac{1}{x} \), we need to consider the values that \( f(x) \) can take as \( x \) varies over all real numbers except zero. The function \( f(x) = \frac{1}{x} \) is undefined at \( x = 0 \) because division by zero is not possible. However, for any other real number \( x \), the function will produce a real number output.

When \( x \) is a positive number, \( \frac{1}{x} \) is also positive and can be any positive real number. As \( x \) gets larger, \( \frac{1}{x} \) gets smaller, approaching 0 but never actually reaching it. Conversely, as \( x \) gets smaller (but still positive), \( \frac{1}{x} \) gets larger without bound.

When \( x \) is a negative number, \( \frac{1}{x} \) is also negative and can be any negative real number. As \( x \) gets more negative (i.e., its absolute value increases), \( \frac{1}{x} \) approaches 0 from the negative side but never actually reaches it. Similarly, as \( x \) gets closer to zero from the negative side, \( \frac{1}{x} \) becomes a large negative number.

Therefore, the function \( f(x) = \frac{1}{x} \) can take any real number value except 0. This means the range of the function is all real numbers except 0.