1.10 Rational functions and holes
What does "multiplicity of a real zero" mean in a rational function?
In the rational function r(x) = \dfrac{(x-3)^2}{(x-3)(x-5)}, is there a hole in the function? If yes, at what value does it occur on the x-axis?
For the rational function r(x) = \dfrac{(x+2)^3}{(x+2)^2(x-4)}, what is the multiplicity of the real zero at x = -2? Does the function have a hole at x = -2?
Does the graph of the rational function r(x) = \dfrac{(x-1)(x-3)}{(x-1)(x+1)}, have a hole? If yes, what are the coordinates of the hole?
What does the mathematical notation lim_(x→c) r(x) = L represent in relation to rational functions?
What is a hole in a rational function? Is it a point on the graph or a point missing from the graph?
True or False: A hole in a rational function occurs when there is a common factor in both the numerator and the denominator that can be divide out.
Determine whether each of the following rational functions has a hole.
Evaluate the limit of the rational functions as xapproaches the given value:
f \left( x \right) = \dfrac{x^2 - 25}{x - 5}, \, x \to 5
f \left( x \right) = \dfrac{x^3 - 8}{x - 2}, \, x \to 2
f \left( x \right) = \dfrac{2x^2 - 18x + 36}{x - 3}, \, x \to 3
f \left( x \right) = \dfrac{x^3 - x^2 - 6x}{x^2 - 4}, \, x \to 2
For the rational function h \left( x \right) = \dfrac{x^3 - 8x}{x^2 - 4x}:
Identify any holes in the graph of h \left( x \right) and their corresponding x values.
For each hole identified in part (a), evaluate the limit of h \left( x \right) as x approaches the hole.
Identify any vertical asymptotes in the graph of h \left( x \right) and their corresponding x values.
For each rational function:
Find the x-value where there is a hole.
Predict the y-value of the hole as x approaches that value from both directions.
Identify the hole(s) in the graph and provide the corresponding (x, y) coordinates of the hole(s).
For each rational function:
Without cancelling, rewrite the function in factorised form.
Identify any holes of the function.
State the domain of the function using interval notation.
Here is the graph of the rational function h(x) = \dfrac{x^3 - 2x^2 - 5x + 6}{x^2 - 4}.
Identify the holes in the graph.
Rewrite the function in a form that clearly shows the holes.
Given the rational function g\left(x\right)=\dfrac{x^3-8x}{x^2-4}.
Identify the real zeros in the denominator.
Identify the real zeros in the numerator.
Find the holes.
Specify the location of the holes.
Explain the role of the real zeros in the numerator and denominator in determining the holes.
For the rational function f(x)=\dfrac{x^2(x-1)(x+2)^2}{(x-1)(x+2)}:
Identify the real zeros of the function and their multiplicities.
Determine the holes of the function.
Explain how the multiplicity of the real zeros in the numerator and denominator affects the presence of holes.
Here are graphs of two rational functions:
Function 1:
Function 2:
Identify the x-value of the hole for each function.
Determine the limit of each function as x approaches the hole.
Describe how the slopes of the functions are similar or different near their respective holes.
Based on the limits calculated, describe the behavior of each function as x approaches the hole. Are there any differences in their behaviors?
The speed of a vehicle is modeled by the rational function S(t)=\dfrac{3t}{t-2}, where t is the time in seconds. What happens to the speed of the vehicle when t=2 seconds? Explain your answer.
The concentration of a chemical in a solution is modeled by the rational function C(x)=\dfrac{5x^2-20x}{x^2-4x}, where x is the volume of the solution in liters. Analyze the behavior of the concentration as the volume approaches 4 liters. Explain the implications of your findings.
The population of a town is modeled by the rational function P(y)=\dfrac{8y^3-64y^2}{y^3-8y^2}, where y is the number of years since the town was established. What is the population in the town after 8 years? Explain any peculiarities in the population function.
The graph represents the rational function s(x) = \dfrac{(x - 2)(x^2 - 1)}{(x - 2)(x + 3)}:
Identify the hole in the graph.
Identify the common factor(s) in the numerator and denominator of the rational function that cause the hole.
Explain what happens to the common factor(s) you identified in the previous part when you simplify the rational function, and how this relates to the hole in the graph.
For the rational functions:
h(x) = \dfrac{x^3 - 8}{x^2 - 4}
k(x) = \dfrac{x^3 - 27}{x^2 - 9}
Determine the holes of both functions h(x) and k(x). What are the x-coordinates of the holes?
Determine the simplified expressions for h(x) and k(x). What are the expressions for the functions near their respective holes?
Use the simplified expressions to find the limit of each function as x approaches the x-coordinate of their respective holes. What are the limits of h(x) and k(x) as x approaches the holes?
Based on the limits, are the behaviors of the functions near their holes similar or different? Explain your answer.
Two companies are analyzing their profit over a period of time using rational functions. Company A's profit is modeled by the function f(x) = \dfrac{x^2 - 4x + 3}{x - 3}, and Company B's profit is modeled by the function g(x) = \dfrac{x^2 - 5x + 6}{x - 2}. Rewrite them in a form that clearly shows the holes and compare their behavior near the holes. Explain how it affects the profit analysis for each company.