Question

Complete the following statement.
The domain of the reciprocal​ function, f left parenthesis x right parenthesis equals StartFraction 1 Over x EndFraction
​, is​ ______, and its range is​ ______.
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Part 1
The domain of the reciprocal​ function, f left parenthesis x right parenthesis equals StartFraction 1 Over x EndFraction
​, is
  
enter your response here​, and its range is
  
enter your response here.
​(Type your answers in interval​ notation.)

Answer 1

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
\[
(-\infty, 0) \cup (0, \infty).
\]