Given the function:

$f(x) = \frac{1}{x} - 1$

Domain:

The domain of a function consists of all real numbers $x$ for which the function is defined.

• The function $f(x)$ contains a fraction with $x$ in the denominator.

• A function is undefined when the denominator is zero, so we set:

  $x \neq 0$

• Therefore, the domain is all real numbers except $x = 0$.

Domain in notation:

$(-\infty, 0) \cup (0, \infty)$

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Range:

The range consists of all possible values of $f(x)$.

• The function can be rewritten as:

  $f(x) = \frac{1}{x} - 1$

• The key observation is that $f(x)$ never attains $-1$.

  • If $x \to \infty$, then $\frac{1}{x} \to 0$ and $f(x) \to -1$ from above.
  • If $x \to -\infty$, then $\frac{1}{x} \to 0$ and $f(x) \to -1$ from above.
  • If $x \to 0^+$, then $\frac{1}{x} \to \infty$ and $f(x) \to \infty$.
  • If $x \to 0^-$, then $\frac{1}{x} \to -\infty$ and $f(x) \to -\infty$.

Thus, the function can take any real value except $-1$.

Range in notation:

$(-\infty, -1) \cup (-1, \infty)$

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Final Answer:

• Domain: $(-\infty, 0) \cup (0, \infty)$
• Range: $(-\infty, -1) \cup (-1, \infty)$

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Do you want a graphical explanation or more details?

Here are some related questions to deepen understanding:

1. How does the function behave near $x = 0$?
2. What is the horizontal asymptote of $f(x)$?
3. What is the vertical asymptote of $f(x)$?
4. How would shifting the function change its domain and range?
5. How do you find the inverse of $f(x)$?

Tip:

For rational functions, always check for asymptotes to determine restrictions on the domain and range! 🚀