Answer
The function \( y = \frac{3}{x} \) has:
- Domain: All real numbers except \( x = 0 \),
- Vertical asymptote at \( x = 0 \),
- Horizontal asymptote at \( y = 0 \),
- No \( x \)- or \( y \)-intercepts,
- Two branches in the first and third quadrants forming a hyperbola.
Solution
**Step 1. Write the equation**
We are given the function
\[
y = \frac{3}{x}.
\]
**Step 2. Identify the domain**
Since division by zero is undefined, we must have
\[
x \neq 0.
\]
Thus, the domain of the function is all real numbers except \(x = 0\).
**Step 3. Determine the vertical asymptote**
Because the function is undefined at \(x = 0\), there is a vertical asymptote at
\[
x = 0.
\]
**Step 4. Determine the horizontal asymptote**
Rewrite the function as
\[
y = \frac{3}{x}.
\]
For large values (in absolute value) of \(x\), \(y\) approaches 0. Hence, there is a horizontal asymptote at
\[
y = 0.
\]
**Step 5. Find intercepts**
- **\(y\)-intercept:** To find the \(y\)-intercept, set \(x = 0\). However, \(x = 0\) is not in the domain, so there is no \(y\)-intercept.
- **\(x\)-intercept:** To find the \(x\)-intercept, set \(y = 0\):
\[
0 = \frac{3}{x}.
\]
Since \(\frac{3}{x}\) can never equal 0 for any finite \(x\), there is no \(x\)-intercept.
**Step 6. Analyze the behavior of the function**
The function \(y = \frac{3}{x}\) represents a hyperbola. It has two branches:
- For \(x > 0\), \(y\) is positive.
- For \(x < 0\), \(y\) is negative.
**Step 7. Plot some key points**
To understand the shape of the graph, consider a few sample values for \(x\):
- If \(x = 1\), then \(y = \frac{3}{1} = 3\).
- If \(x = 3\), then \(y = \frac{3}{3} = 1\).
- If \(x = -1\), then \(y = \frac{3}{-1} = -3\).
- If \(x = -3\), then \(y = \frac{3}{-3} = -1\).
These points confirm that the branch in the first quadrant (when \(x > 0\) and \(y > 0\)) continuously approaches the horizontal asymptote \(y = 0\) as \(x\) increases and the vertical asymptote \(x = 0\) as \(x\) decreases, and similarly for the branch in the third quadrant.
**Summary**
The function
\[
y = \frac{3}{x}
\]
has:
- Domain: \(x \in \mathbb{R} \setminus \{0\}\),
- Vertical asymptote: \(x = 0\),
- Horizontal asymptote: \(y = 0\),
- No \(x\)- or \(y\)-intercepts,
- Two branches (one in the first quadrant and one in the third quadrant) representing a hyperbola.
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