Question
Graph the equation. Describe the domain and range. 15. \( y=\frac{1}{x} \)
Ask by Montgomery Brewer. in the United States
Mar 19,2025
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Answer
The graph of \( y = \frac{1}{x} \) is a hyperbola with a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \). The domain is all real numbers except \( x = 0 \), and the range is all real numbers except \( y = 0 \).
Solution
1. **Given Equation:**
\[
y = \frac{1}{x}
\]
2. **Graph Description:**
- The graph of \( y=\frac{1}{x} \) is a hyperbola.
- There is a vertical asymptote at \( x = 0 \) (the graph is not defined at \( x=0 \)).
- There is a horizontal asymptote at \( y = 0 \) as \( x \) approaches positive or negative infinity.
- The hyperbola has two branches: one in the first quadrant (where both \( x \) and \( y \) are positive) and one in the third quadrant (where both \( x \) and \( y \) are negative).
3. **Domain:**
- The domain consists of all real numbers except where the function is undefined.
- Since division by zero is undefined, \( x=0 \) is excluded.
\[
\text{Domain: } \{ x \in \mathbb{R} \mid x \neq 0 \}
\]
4. **Range:**
- The range is all the possible \( y \)-values that the function can take.
- As there is no \( y \) corresponding to \( x=0 \), and the output never equals 0, \( y \) can be any real number except 0.
\[
\text{Range: } \{ y \in \mathbb{R} \mid y \neq 0 \}
\]
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The graph of \( y=\frac{1}{x} \) is a hyperbola that approaches the x-axis and y-axis but never touches them, resulting in two separate curves in the first and third quadrants. As x approaches zero from the right, y approaches infinity, and as x approaches zero from the left, y approaches negative infinity. For the domain, all real numbers are allowed except for zero (since you can't divide by zero), so it’s (-∞, 0) ∪ (0, ∞). The range is also all real numbers except for zero, represented as (-∞, 0) ∪ (0, ∞). This creates a fascinating function where both variables can’t be zero simultaneously!
