Rational Functions - Unit 2 Test

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Questions and Answers

In the inequality $\frac{40}{t+4} - 1 > 0$, what is the significance of making the denominator common when solving for $t$?

Making the denominator common allows us to combine the terms into a single fraction, which is essential for determining the intervals where the inequality holds true.

Given a rational function with a vertical asymptote at $x = -1$ and an x-intercept at $x = 2$, explain why the denominator must contain the factor $(x + 1)$ and the numerator must contain the factor $(x - 2)$.

A vertical asymptote at $x = -1$ implies the function is undefined at that point, making $(x + 1)$ a factor of the denominator. An x-intercept at $x = 2$ means the function equals zero at that point, making $(x - 2)$ a factor of the numerator.

How does the horizontal asymptote at $y = -\frac{1}{2}$ influence the coefficients of the numerator and denominator in the rational function $f(x) = -\frac{1}{2} \cdot \frac{x-2}{x+1}$?

The horizontal asymptote at $y = -\frac{1}{2}$ indicates that the ratio of the leading coefficients of the numerator and denominator is equal to $-\frac{1}{2}$.

When solving the inequality $\frac{t-36}{t+4} < 0$, why is it important to consider both the roots of the numerator and the denominator?

<p>The roots of both the numerator and denominator are critical points that divide the number line into intervals. The sign of the rational expression can only change at these points, so we need to test each interval to determine where the inequality holds.</p> Signup and view all the answers

Is it possible to have another function with the same key features (vertical asymptote, x-intercept, horizontal asymptote) as $f(x) = -\frac{1}{2} \cdot \frac{x-2}{x+1}$? Explain your reasoning.

<p>No, it is not possible to have another function with those key features. Once the positions of the key features are determined and the ratio of the coefficients is determined, the form of the rational function is fixed.</p> Signup and view all the answers

Flashcards

Vertical Asymptote

A line x = a where a function approaches infinity or negative infinity.

X-intercept

The x-value where the function crosses the x-axis, f(x) = 0.

Horizontal Asymptote

A line y = b that a function approaches as x goes to infinity.

Domain of a function

The set of all possible input values (x) for which the function is defined.

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Function Equation Formation

Creating a rational function based on key features like asymptotes and intercepts.

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Study Notes

Rational Functions - Unit 2 Test

General Marking Notations:
- M: Marks for method
- A: Marks for application
- R: Marks for reasoning
- N: Marks awarded when no working is shown, but the answer is correct (no M, A, or R marks)

Question 1

Part a):
- Given inequality: (3 / (x - 4)) < 5
- Making the denominator common: (-5x + 23) / (x - 4) < 0
- Simplifying: (x-23/5) / (x-4) <0
- Critical values: x = 4, x = 23/5.
- Solution: x ∈ (-∞, 4) U (23/5, ∞).
Part b):
- Given equation: (x² - 8x + 15) / (x² + 5x + 4) ≥ 0
- Factorising: ((x-3)(x-5)) / ((x + 4)(x + 1)) ≥ 0
- Critical values: x = -4, x = -1, x = 3, x = 5.
- Solution set: x ∈ (-∞, -4) U (-1, 3] U [5, ∞)

Question 2.A

R(t) = (2t) / (t² + 4t):
- Rate is non-zero for all t
- Limit as t approaches ∞: 0.
- Horizontal Asymptote: y = 0
- Vertical asymptote: t = -4
- Behavior around asymptote: R(-4⁻) →-∞ and R(-4⁺) → ∞.

Question 2.B

Horizontal Asymptote: R(t) = 0

Question 2.C

Rate greater than 0.05 g/s: (2t) / (t² + 4t) > 0.05
Finding the values for t where the inequality holds. A solution for t is given, between 0 to 36
- Domain: t ∈ (0, 36)

Question 3

Part a):
- Vertical Asymptote at x = -1, so (x + 1) is in the denominator
- x-intercept at x = 2, so (x - 2) is in the numerator
- Horizontal Asymptote at y = 1/2, Meaning that the numerator coefficient divided by the denominator coefficient is 1/2
- Function: f(x) = (x - 2) / (2(x + 1))
Part b):
- Yes, it is possible to have a function with similar characteristics by changing the variable restriction.

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Description

This quiz focuses on rational functions, including solving inequalities and equations. Students will demonstrate their understanding of critical values, solution sets, and asymptotic behavior through a series of mathematical problems. Perfect for assessing knowledge within this essential topic in algebra.