📝 Chapter 3: Rational Functions

For f(x) = 1/(x²-9): a) State the domain b) Find all asymptotes c) Determine the behaviour near each VA

For g(x) = (3x+2)/(x-1): a) State VA and HA b) Find intercepts c) Sketch the graph

Identify the hole in h(x) = (x²-4)/(x²+x-6). State its coordinates.

Solve: 2/(x+1) + 3/(x-2) = 1. Check for extraneous solutions.

Solve and graph the solution: (x-3)/(x+1) < 2

Design a rational function with EXACTLY the following features: VA at x=-1 and x=4, HA at y=2, x-intercept at (3,0), and a hole at x=0. Write the equation and verify each feature.

A student says "a rational function can never cross its horizontal asymptote." Find a counterexample and explain why the student is wrong.

Explain why f(x) = (x²+1)/(x²-1) has no x-intercepts but g(x) = (x²-1)/(x²+1) has two. What does this tell you about the relationship between numerator and denominator?

Investigate: For f(x) = (ax²+bx+c)/(dx+e), when does an oblique asymptote occur? Find the oblique asymptote of f(x) = (2x²+3x-1)/(x+1) using polynomial long division.

Write a step-by-step guide for sketching any rational function of the form f(x) = (linear)/(linear). Include: asymptotes, intercepts, test points, and behaviour near asymptotes.

Explain the concept of a "hole" in a rational function. What causes it? How do you find its location? How does it differ from a vertical asymptote? Use at least one specific example.

Compare and contrast the graphs of f(x) = 1/x, g(x) = 1/x², and h(x) = 1/(x-2)+3. Create a table or chart showing their key features side by side.

A classmate solved 5/x = 2/(x-3) and got x = 5. They substituted back and confirmed it works. Write feedback explaining whether their process was mathematically rigorous and what step they should never skip.

A school is planning a field trip that costs $400 for the bus plus $15 per student. a) Write a rational function C(n) for the cost per student. b) Graph C(n). c) How many students are needed so cost per student is under $25?

The concentration of a drug in the bloodstream is modeled by C(t) = 50t/(t²+4) mg/L, where t is hours after taking the drug. a) What is the concentration after 2 hours? b) When does concentration drop below 5 mg/L? c) What happens to concentration as t→∞ and what does this mean medically?

Two pipes fill a pool. Pipe A fills it in 6 hours alone, Pipe B in 8 hours alone. A drain empties it in 12 hours. If all three are open, how long to fill the pool? Set up and solve using rational equations.

A ferry company charges $10/person and gets 200 passengers daily. Market research shows each $0.50 increase loses 5 passengers. a) Write a rational function for revenue per passenger as a function of price increases. b) Is there a price that gives $0 revenue? What does that mean?