The reciprocal function is defined as \( f(x) = \frac{1}{x} \).
Domain: The domain of the reciprocal function is all real numbers except for \( x = 0 \) because division by zero is undefined. In interval notation, this is expressed as: \[ (-\infty, 0) \cup (0, \infty) \]
Range: The range of the reciprocal function is also all real numbers except for \( y = 0 \) because the function can never equal zero (as \( f(x) \) approaches \( 0 \) as \( x \) approaches infinity or negative infinity, but never actually reaches it). In interval notation, the range is: \[ (-\infty, 0) \cup (0, \infty) \]
So the completed statement would be: The domain of the reciprocal function, \( f(x) = \frac{1}{x} \), is \[ (-\infty, 0) \cup (0, \infty) \] and its range is \[ (-\infty, 0) \cup (0, \infty). \]
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