AP Pre Calculus Rational Functions

Liam, Grace, and Harper are studying functions in their math class. Their teacher asks them to define a rational function in terms of polynomial functions. How should they respond?

• A rational function is a function that can be written as the product of two polynomial functions.

• A rational function is a function that can be written as the difference of two polynomial functions.

• A rational function is a function that can be written as the sum of two polynomial functions.

• A rational function is a function that can be written as the quotient of two polynomial functions, where the denominator is not equal to zero.

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Oliver and Daniel are working on a project to model the water flow through a pipe system using a rational function. How do they determine the domain of this function?

• Find the values of x that make the numerator non-zero.

• Find the values of x that make the denominator zero.

• Find the values of x that make the function undefined.

• Find the values of x that make the denominator non-zero.

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Lily, Michael, and Benjamin are studying rational functions in their math class. Their teacher asks them to identify an asymptote in a rational function. How would they describe it?

• A point where the graph changes direction.

• A line that the graph approaches but never touches.

• A line that the graph intersects at multiple points.

• A line that the graph touches at one point.

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Luna, Elijah, and Noah are trying to design a roller coaster. They need to calculate the points where the roller coaster track would go infinitely high (vertical asymptotes). How should they do this for a rational function representing the track's shape?

• They should take the derivative of the function and find the x-values where the derivative is undefined.

• They should set the numerator of the function equal to zero and solve for x.

• They should set the denominator of the function equal to zero and solve for x.

• They should set both the numerator and denominator of the function equal to zero and solve for x.

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Lily, Evelyn, and Scarlett are studying rational functions in their math class. They come across a problem that asks, 'What is the difference between a removable and non-removable discontinuity in a rational function?' How would you explain this to them?

• A removable discontinuity can be filled by removing a value from the function at that point, like erasing a pencil mark on a graph, while a non-removable discontinuity cannot be filled by removing a value from the function at that point, like a permanent marker line on a graph.

• A removable discontinuity occurs when the function is undefined at a certain point, like a hole in a road, while a non-removable discontinuity occurs when the function is defined at a certain point, like a bridge over a river.

• A removable discontinuity can be fixed by changing the function's equation, like fixing a broken toy with a new part, while a non-removable discontinuity cannot be fixed by changing the function's equation, like a shattered vase that can't be put back together.

• A removable discontinuity can be filled by assigning a value to the function at that point, like filling a hole in a wall with plaster, while a non-removable discontinuity cannot be filled by assigning a value to the function at that point, like a window in a wall that can't be filled with plaster.

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Michael, David, and William are studying rational functions. They are trying to understand the end behavior of these functions. How is the end behavior of a rational function determined?

• Michael suggests that the end behavior of a rational function is determined by the sum of the degrees of the numerator and denominator.

• David thinks that the end behavior of a rational function is determined by the constant term of the numerator and denominator.

• William believes that the end behavior of a rational function is determined by the leading coefficient of the numerator and denominator.

• They read in a book that the end behavior of a rational function is determined by the degrees of the numerator and denominator.

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Abigail, Nora, and Elijah are studying rational functions in their math class. They are trying to find the horizontal asymptotes of a rational function in a real-world problem. How should they approach this task?

• They should look at the degrees of the numerator and denominator.

• They should calculate the derivative of the function.

• They should find the average of the numerator and denominator.

• They should look at the coefficients of the numerator and denominator.

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Maya, Avery, and Sophia are studying the end behavior of rational functions in their math class. What are the possible types of end behavior for a rational function they might encounter?

• Approaching zero

• Approaching a constant value

• Approaching a finite number

• Approaching positive infinity or approaching negative infinity

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Arjun is trying to find the x-intercepts of a rational function that models the profit of a company. How should he determine the x-intercepts of this function?

• Set the denominator equal to zero and solve for x.

• Set the whole function equal to zero and solve for x.

• Take the derivative of the function and find the x-values where the derivative is zero.

• Set the numerator equal to zero and solve for x.

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Select all the following statements that are true.

• Vertical asymptote at x=1

• Vertical asymptote at x=0

• No vertical asymptote

• Hole at (-1, 0)

• Hole at (1, -1)

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