Function Families — Hyperbola y=1/x
Function Families — Hyperbola y=1/x. Practice questions to deepen understanding of the hyperbola y=1/x. Online math practice with full solutions and step-by-step explanations.
Hyperbola y=1/x practice — vertical and horizontal asymptotes, sign in the quadrants, behavior at the ends. Getting to know the rational function.
Asymptotes, sign.
Question 1
10.00 pts
➗ Hyperbola:
What is the basic form of a hyperbola?
Explanation:
| ➗ Hyperbola The basic form: \(f(x) = \frac{1}{x}\) or: \(f(x) = x^{-1}\) Name: Called "hyperbola" or "the simple rational function" Special property: It has two separate "branches"! One in the first quadrant ↗ One in the third quadrant ↙ |
Question 2
10.00 pts
📊 domain/interval:
the domain of definition \(f(x) = \frac{1}{x}\)?
Explanation:
| Explanation: The correct answer follows from the Hebrew source using the same mathematical rule and the same formulas. Read the given expression, preserve the signs and formulas exactly, and apply the stated rule step by step. The selected answer is the one that matches the calculation in the original Hebrew item. |
Question 3
10.00 pts
📈 range:
range \(f(x) = \frac{1}{x}\)?
Explanation:
| Explanation: The correct answer follows from the Hebrew source using the same mathematical rule and the same formulas. Read the given expression, preserve the signs and formulas exactly, and apply the stated rule step by step. The selected answer is the one that matches the calculation in the original Hebrew item. |
Question 4
10.00 pts
⬆️➡️ Asymptotes:
What asymptotes does \(f(x) = \frac{1}{x}\) have?
Explanation:
| ⬆️➡️ Asymptotes of the hyperbola Two asymptotes: 1️⃣ Vertical: \(x=0\) (the y-axis) 2️⃣ Horizontal: \(y=0\) (the x-axis) Why? Vertical at \(x=0\): Because \(f(0)\) is undefined The graph "escapes" to infinity near 0 Horizontal at \(y=0\): As \(x \to \pm\infty\): \(\frac{1}{x} \to 0\) The axes are the asymptotes! This is a unique property of \(\frac{1}{x}\) |
Question 5
10.00 pts
📉 Monotonicity:
What is the monotonicity of \(f(x) = \frac{{1}}{{x}}\)?
Explanation:
| Rule: \(f(x)=\frac{{1}}{{x}}\) is decreasing on each interval separately: \((-\infty,0)\) and \((0,\infty)\). Why? As \(x\) increases (within each interval), the denominator increases → the fraction decreases. Example (positive side): \(f(1)=1, f(2)=0.5, f(10)=0.1\) — decreasing. Important: Not decreasing "throughout its entire domain" — this would require comparing values across the discontinuity at \(x=0\), which is invalid. |
Question 6
10.00 pts
➕➖ Sign:
Where is \(f(x) = \frac{1}{x}\) positive and where is it negative?
Explanation:
| ➕➖ Sign of the hyperbola The simple rule: \(\frac{1}{x}\) preserves the sign! • \(x > 0\) → \(\frac{1}{x} > 0\) ✓ • \(x < 0\) → \(\frac{1}{x} < 0\) ✓ Positive: On \((0, \infty)\) Every positive \(x\) → \(\frac{1}{x}\) positive Examples: \(f(2) = \frac{1}{2} = 0.5 > 0\) ✓ \(f(5) = \frac{1}{5} = 0.2 > 0\) ✓ Negative: On \((-\infty, 0)\) Every negative \(x\) → \(\frac{1}{x}\) negative Examples: \(f(-2) = \frac{1}{-2} = -0.5 < 0\) ✓ \(f(-5) = \frac{1}{-5} = -0.2 < 0\) ✓ Never zero! Because \(\frac{1}{x} \neq 0\) for every \(x\) |
Question 7
10.00 pts
📍 Special points:
What important points does the graph of \(f(x) = \frac{1}{x}\) pass through?
Explanation:
| 📍 Reference points Two main points: 1️⃣ (1, 1) \(\frac{1}{1} = 1\) 2️⃣ (-1, -1) \(\frac{1}{-1} = -1\) Why are they important? These are the points where \(x = y\) Reference points for sketching the graph Other useful points: \((2, 0.5)\): \(\frac{1}{2}\) \((0.5, 2)\): \(\frac{1}{0.5} = 2\) \((-2, -0.5)\): \(\frac{1}{-2}\) \((-0.5, -2)\): \(\frac{1}{-0.5} = -2\) Note: There are no intersections with the axes! (because both axes are asymptotes) |
Question 8
10.00 pts
🔄 Symmetry:
What symmetry does \(f(x) = \frac{1}{x}\) have?
Explanation:
| 🔄 Symmetry - odd function The property: \(f(-x) = -f(x)\) Odd function Symmetry with respect to the origin! Proof: \(f(x) = \frac{1}{x}\) \(f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x)\) ✓ What does this mean? If you rotate the graph 180° around the origin (0,0) you get the same graph! Example: \(f(2) = 0.5\) \(f(-2) = -0.5\) Opposite values! ✓ |
Question 9
10.00 pts
⚠️ Common mistake:
A student said: "The hyperbola is decreasing on the entire domain". What is the mistake?
Explanation:
❌ A confusing mistake! Two separate intervals! The problem: What the student thought: "At every point I check, the function is decreasing so it is decreasing on the entire domain" ❌ Not exactly! ✓ The correct view: There are two separate intervals: 1️⃣ \((-\infty, 0)\): decreasing ↘ 2️⃣ \((0, \infty)\): decreasing ↘ But there is a "jump" at \(x=0\)! Why does it matter? Because if you compare values from the two sides: \(f(-1) = -1\) \(f(1) = 1\) \(-1 < 1\) means an increase! Because of the jump at 0 The correct phrasing: "Decreasing on each connected interval of its domain" or: "Decreasing on \((-\infty, 0)\) and on \((0, \infty)\)" |
Question 10
10.00 pts
📚 Summary:
What is the unique property of the hyperbola?
Explanation:
📚 Summary - hyperbola ➗ The form: \(f(x) = \frac{1}{x}\) 📊 Domain and range: Domain: \(\mathbb{R} \setminus \{0\}\) Range: \(\mathbb{R} \setminus \{0\}\) Both exclude 0! ⬆️➡️ Asymptotes: Vertical: \(x=0\) (the y-axis) Horizontal: \(y=0\) (the x-axis) The axes are the asymptotes! 📉 Monotonicity: Decreasing on \((-\infty, 0)\) ↘ Decreasing on \((0, \infty)\) ↘ On each interval separately! ➕➖ Sign: Positive: \((0, \infty)\) Negative: \((-\infty, 0)\) Sign-preserving! 🔄 Symmetry: Odd function \(f(-x) = -f(x)\) Symmetry with respect to the origin ✨ Unique: Two separate branches The axes as asymptotes Symmetry with respect to the origin |