Functions $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$ and Their Graphs
Big Idea: These two functions belong to a special family called reciprocal functions. Unlike linear or quadratic functions, they produce curves called hyperbolas that never touch the axes. Understanding them opens the door to rational functions — one of the most powerful tools in algebra.
Function 1: $y = \frac{1}{x}$
Definition
For every input $x$ (except $x = 0$), the output is the reciprocal of $x$.
Key Properties
- Domain: All real numbers except $x = 0$, i.e., $x \in (-\infty, 0) \cup (0, +\infty)$
- Range: All real numbers except $y = 0$, i.e., $y \in (-\infty, 0) \cup (0, +\infty)$
- Asymptotes: The $x$-axis ($y = 0$) is a horizontal asymptote; the $y$-axis ($x = 0$) is a vertical asymptote
- Symmetry: The graph has odd symmetry — it is symmetric about the origin (rotational symmetry of 180°)
- Quadrants: The curve lies in Quadrant I and Quadrant III
- Behavior: As $x \to +\infty$, $y \to 0^+$. As $x \to 0^+$, $y \to +\infty$
Table of Values
| $x$ | $-4$ | $-2$ | $-1$ | $-0.5$ | $0.5$ | $1$ | $2$ | $4$ |
|---|---|---|---|---|---|---|---|---|
| $y$ | $-0.25$ | $-0.5$ | $-1$ | $-2$ | $2$ | $1$ | $0.5$ | $0.25$ |
Function 2: $y = \frac{1}{x^2}$
Definition
For every input $x$ (except $x = 0$), the output is the reciprocal of $x$ squared.
Key Properties
- Domain: All real numbers except $x = 0$, i.e., $x \in (-\infty, 0) \cup (0, +\infty)$
- Range: All positive real numbers, i.e., $y \in (0, +\infty)$
- Asymptotes: The $x$-axis ($y = 0$) is a horizontal asymptote; the $y$-axis ($x = 0$) is a vertical asymptote
- Symmetry: The graph has even symmetry — it is symmetric about the $y$-axis (mirror symmetry)
- Quadrants: The curve lies in Quadrant I and Quadrant II only (always positive!)
- Behavior: As $x \to +\infty$, $y \to 0^+$. As $x \to 0^+$, $y \to +\infty$
Table of Values
| $x$ | $-4$ | $-2$ | $-1$ | $-0.5$ | $0.5$ | $1$ | $2$ | $4$ |
|---|---|---|---|---|---|---|---|---|
| $y$ | $0.0625$ | $0.25$ | $1$ | $4$ | $4$ | $1$ | $0.25$ | $0.0625$ |