The reciprocal function, typically represented as $f(x) = \frac{1}{x}$, has a unique set of characteristics that define its behavior and graph:

1. Domain and Range

• Domain: All real numbers except $x = 0$, because dividing by zero is undefined.
• Range: All real numbers except $y = 0$, since $\frac{1}{x}$ never equals zero for any non-zero $x$.

2. Asymptotes

• Vertical Asymptote: The line $x = 0$ is a vertical asymptote, as the function values become infinitely large as $x$ approaches zero from the positive or negative direction.
• Horizontal Asymptote: The line $y = 0$ is a horizontal asymptote, since as $x$ becomes very large (positive or negative), the values of $\frac{1}{x}$ approach zero but never actually reach it.

3. Symmetry

• The reciprocal function is odd (symmetric about the origin). This means that $f(-x) = -f(x)$, which produces a symmetric graph about the origin.

4. Intervals of Increase and Decrease

• The function decreases throughout its domain:
  • For $x > 0$, $f(x) = \frac{1}{x}$ decreases as $x$ increases.
  • For $x < 0$, $f(x) = \frac{1}{x}$ also decreases as $x$ becomes more negative.

5. Behavior Near the Asymptotes

• As $x$ approaches zero from the right ($x \to 0^+$), $f(x) \to +\infty$.
• As $x$ approaches zero from the left ($x \to 0^-$), $f(x) \to -\infty$.

Graph of $f(x) = \frac{1}{x}$:

The graph of the reciprocal function has two branches, one in the first quadrant (for positive $x$) and one in the third quadrant (for negative $x$), both approaching the asymptotes without crossing them.

Would you like more details on any specific property of the reciprocal function?

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Here are some additional questions related to the reciprocal function:

1. How does the reciprocal function transform under vertical or horizontal scaling?
2. What happens to the graph of the reciprocal function when it is shifted left or right?
3. How does the reciprocal function behave under different transformations, such as adding a constant to $f(x) = \frac{1}{x}$?
4. How do other types of reciprocal functions, like $f(x) = \frac{1}{x^2}$, differ in properties?
5. What are the applications of the reciprocal function in real-world contexts?

Tip: Visualizing the reciprocal function’s graph can make understanding its asymptotic behavior clearer.