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Questions and Answers
What is the outcome when substituting $x = 0$ into the inequality $2x^2 - 3x + 1 ≤ 0$?
What is the outcome when substituting $x = 0$ into the inequality $2x^2 - 3x + 1 ≤ 0$?
Which test point correctly supports the interval $(-1/2, 1)$ solving the inequality?
Which test point correctly supports the interval $(-1/2, 1)$ solving the inequality?
What does the tested point $x = 2$ reveal about the interval $(1, ext{∞})$?
What does the tested point $x = 2$ reveal about the interval $(1, ext{∞})$?
What is the correct solution set for the inequality $2x^2 - 3x + 1 ≤ 0$ based on tested intervals?
What is the correct solution set for the inequality $2x^2 - 3x + 1 ≤ 0$ based on tested intervals?
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In solving $2x^2 - 3x + 1 ≤ 0$, which conclusion can be drawn from testing $x = -1/2$?
In solving $2x^2 - 3x + 1 ≤ 0$, which conclusion can be drawn from testing $x = -1/2$?
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What defines the domain of a function?
What defines the domain of a function?
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Which of the following statements is true regarding a function?
Which of the following statements is true regarding a function?
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What happens in computing when a division by zero occurs?
What happens in computing when a division by zero occurs?
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In the context of functions, what does the range represent?
In the context of functions, what does the range represent?
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Why is dividing by zero prohibited in mathematics?
Why is dividing by zero prohibited in mathematics?
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What is the domain of the function $g(x) = -3x^2 + 4x + 5$?
What is the domain of the function $g(x) = -3x^2 + 4x + 5$?
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For the function $f(x) = \frac{x-4}{x+3}$, what value must be excluded from the domain?
For the function $f(x) = \frac{x-4}{x+3}$, what value must be excluded from the domain?
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What is the domain of the function $f(x) = \frac{x+1}{(x-3)(x+5)}$?
What is the domain of the function $f(x) = \frac{x+1}{(x-3)(x+5)}$?
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What is the domain of the function $f(x) = \frac{4}{x^2 - 9}$?
What is the domain of the function $f(x) = \frac{4}{x^2 - 9}$?
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For the function $f(x) = \frac{x-1}{x^2 + 4x + 3}$, which values are excluded from the domain?
For the function $f(x) = \frac{x-1}{x^2 + 4x + 3}$, which values are excluded from the domain?
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What is the domain of the function $f(x) = \frac{5}{x^2 + 9}$?
What is the domain of the function $f(x) = \frac{5}{x^2 + 9}$?
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What values need to be excluded from the domain for the function $f(x) = \frac{x}{x-1}$?
What values need to be excluded from the domain for the function $f(x) = \frac{x}{x-1}$?
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From the graph of the function $f(x) = x + 3$, what is the domain?
From the graph of the function $f(x) = x + 3$, what is the domain?
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What is the domain of the function $f(x) = \frac{1}{x}$?
What is the domain of the function $f(x) = \frac{1}{x}$?
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In the function $f(x) = x^2 + 1$, which of the following is true about its domain?
In the function $f(x) = x^2 + 1$, which of the following is true about its domain?
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What is the solution set for the inequality $3x - 2 + 3 \geq 11$?
What is the solution set for the inequality $3x - 2 + 3 \geq 11$?
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Which of the following describes the equivalent form of the inequality $2x + 1 > 9$?
Which of the following describes the equivalent form of the inequality $2x + 1 > 9$?
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What does the Vertical Line Test assess regarding a graph?
What does the Vertical Line Test assess regarding a graph?
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For the case when $x < a$, where $a > 0$, what is the equivalent inequality?
For the case when $x < a$, where $a > 0$, what is the equivalent inequality?
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What defines the set of rational numbers?
What defines the set of rational numbers?
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What is a condition for a relation to be considered a function?
What is a condition for a relation to be considered a function?
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Identify the solution set for the inequality $2x - 4 > 8$.
Identify the solution set for the inequality $2x - 4 > 8$.
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Which statement is true regarding the inequality $x - 3 < 5$?
Which statement is true regarding the inequality $x - 3 < 5$?
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What is the solution for the inequality $2x + 1 < -9$?
What is the solution for the inequality $2x + 1 < -9$?
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What is the solution set for the inequality $2x + 1 < -9$?
What is the solution set for the inequality $2x + 1 < -9$?
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What technique is used to solve the inequality $2x^2 - 3x + 1 ≤ 0$?
What technique is used to solve the inequality $2x^2 - 3x + 1 ≤ 0$?
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In the context of absolute value inequalities, what does $x > a$ result in?
In the context of absolute value inequalities, what does $x > a$ result in?
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When testing the intervals from the equation $2x^2 - 3x + 1 = 0$, what must be done first?
When testing the intervals from the equation $2x^2 - 3x + 1 = 0$, what must be done first?
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Which of the following sets contains only integers?
Which of the following sets contains only integers?
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Which statement is true about irrational numbers?
Which statement is true about irrational numbers?
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What does the notation $x ∈ (−∞, −5)$ indicate?
What does the notation $x ∈ (−∞, −5)$ indicate?
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Study Notes
MATH100 Precalculus - Lecture Notes
- Lecture #1: Covers introduction to functions, solving inequalities, and domain and range of a function.
Real Numbers
- Real numbers are the combination of rational and irrational numbers.
- Rational numbers (Q): Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Examples: -1/2, 0, 3, 2.5.
- Irrational numbers (Q'): Numbers that cannot be expressed as a fraction of two integers. Examples: π, √2, √3
- Integers (Z): Whole numbers, including zero and negative numbers. Examples: ..., -3, -2, -1, 0, 1, 2, 3,...
- Natural numbers (N): Positive whole numbers. Examples: 1, 2, 3,...
Solving Inequalities: Examples
- Example 1: Solving 2x - 4 > 8 results in x > 6. The solution set is x ∈ (6, ∞).
- Example 2: Solving 2x + 1 < -9 results in x < -5. The solution set is x ∈ (-∞, -5).
- Example 3: Solving 1 - 2x < 9 (needs the solution)
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Example 4: Solving 2x² - 3x + 1 ≤ 0 results in the interval [1/2, 1].
- Finding zeros of 2x² - 3x + 1 = 0: x = 1/2 and x = 1.
- Testing intervals to determine the solution set to the inequality.
Absolute Value Inequalities: Examples
- Case 1: |x| < a is equivalent to −a < x < a
- Case 2: |x| > a is equivalent to x < −a or x > a
- Example 1: Solving |x − 3| < 5 results in −2 < x < 8. The solution set is x ∈ (−2, 8).
- Example 2: Solving |2x + 1| > 9 results in x < −5 or x > 4. The solution set is x ∈ (−∞, −5) ∪ (4, ∞).
- Example 3: Solving |3x − 2| + 3 ≥ 11 results in x ≥ 10/3 or x ≤ −2/3. The solution set is x ∈ (−∞, −2/3] ∪ [10/3, ∞).
Function Definition
- A function is a relation that assigns each input value to exactly one output value.
- The notation is y = f(x), where x is the input, y is the output, and f is the function.
- Examples include y = x², y = √x.
- Function examples are plotted (parabolas, sine waves, circles)
The Vertical Line Test
- A graph represents a function if no vertical line intersects the graph at more than one point.
Domain and Range of a Function
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Domain: The set of all possible input values (x-values) for which the function is defined.
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Range: The set of all possible output values (y-values) that the function can produce.
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Examples:
- Example 1: g(x) = −3x² + 4x + 5 has a domain of all real numbers.
- Example 2: f(x) = (x-4)/(x+3) has a domain of all real numbers except x = -3.
- Example 3: f(x) = (x+1)/((x-3)(x+5)) has a domain of all real numbers except x = 3 and x = -5.
- Example 4: f(x) = 4/(x²−9) has a domain of all real numbers except x = 3 and x = −3
- Example 5: f(x) = (x-1)/(x² + 4x + 3) has a domain of all real numbers except x = −1 and x=−3
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Finding domain and range from graphs: visual analysis
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Why aren't we allowed to divide by zero? A historical example of a real-world problem stemming from attempting to divide by zero in computer code.
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Description
This quiz covers key concepts from MATH100 Precalculus, including an introduction to functions, real numbers, and methods for solving inequalities. Participants will explore types of numbers such as rational, irrational, integers, and natural numbers, along with practical examples of solving inequalities. Prepare to test your knowledge on these foundational topics!
