Session 04-05: Tasks
Rational & Logarithmic Functions
Rational & Logarithmic Functions
Problem 1: Basic Asymptote Identification (x)
For each rational function, identify all vertical and horizontal asymptotes:
\(f(x) = \frac{3}{x-2}\)
\(g(x) = \frac{2x+1}{x-3}\)
\(h(x) = \frac{x^2+1}{x^2-4}\)
\(k(x) = \frac{x^3+2x}{x^2+1}\)
Problem 2: Holes vs Asymptotes (x)
Identify whether each function has a hole or vertical asymptote at the given point:
\(f(x) = \frac{x^2-9}{x-3}\) at \(x = 3\)
\(g(x) = \frac{x^2-4x+4}{x-2}\) at \(x = 2\)
\(h(x) = \frac{x^2-1}{(x-1)(x+2)}\) at \(x = 1\)
Problem 3: Average Cost Analysis (xx)
A company has total cost function \(C(x) = 3600 + 24x + 0.01x^2\) dollars for producing \(x\) units.
- Find the average cost function \(AC(x)\)
- Find the production level that minimizes average cost
- What is the minimum average cost?
- Find \(\lim_{x \to \infty} AC(x)\) and interpret its meaning
Problem 4: Logarithmic Properties (xx)
Simplify each expression using logarithm properties:
\(\log_3(27x^2)\)
\(\ln(e^{2x} \cdot \sqrt{x})\)
\(2\log_5(5x) - \log_5(x^2)\)
\(\log_2(8) + \log_2(x/4)\)
Problem 5: Graph Sketching (xx)
Sketch the rational function \(f(x) = \frac{2x-4}{x+1}\) by finding:
- Domain and asymptotes
- x and y intercepts
- Sign analysis
- End behavior
Problem 6: Logarithmic Equation (xxx)
Solve the equation: \(\log_3(x+8) + \log_3(x) = 2\)
Problem 7: Semi-log Data Analysis (xxx)
A bacteria culture shows the following population data:
| Time (hours) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| Population | 100 | 400 | 1600 | 6400 | 25600 |
- Show that this represents exponential growth
- Find the growth formula \(P(t) = P_0 \cdot b^t\)
- What is the doubling time?
- Predict the population at \(t = 10\) hours
Problem 8: Complex Rational Function (xxxx)
Analyze the function \(f(x) = \frac{x^2 - 4}{x - 1}\) completely:
- Find all asymptotes and holes
- Find the x and y intercepts
- Find where \(f(x) = x + 2\)
- Sketch the function showing all key features