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Questions and Answers
Rational numbers can always be expressed as a fraction $\frac{p}{q}$, where p and q are integers and $q \neq 0$.
Rational numbers can always be expressed as a fraction $\frac{p}{q}$, where p and q are integers and $q \neq 0$.
True (A)
The number $\pi$ (pi) is an example of a rational number.
The number $\pi$ (pi) is an example of a rational number.
False (B)
Solving an inequality is the process of finding the set of all numbers that, when substituted for the variable, make the inequality a true statement.
Solving an inequality is the process of finding the set of all numbers that, when substituted for the variable, make the inequality a true statement.
True (A)
When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol remains the same.
When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol remains the same.
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The domain of the function $f(x) = \sqrt{x}$ includes all real numbers.
The domain of the function $f(x) = \sqrt{x}$ includes all real numbers.
A function is one-to-one if every element of the range corresponds to exactly one element of the domain.
A function is one-to-one if every element of the range corresponds to exactly one element of the domain.
The function $f(x) = x^2$ is a one-to-one function.
The function $f(x) = x^2$ is a one-to-one function.
The inverse of a function, denoted as $f^{-1}(x)$, always exists for any function $f(x)$.
The inverse of a function, denoted as $f^{-1}(x)$, always exists for any function $f(x)$.
If $f(x) = 2x + 1$, then $f^{-1}(x) = \frac{x - 1}{2}$.
If $f(x) = 2x + 1$, then $f^{-1}(x) = \frac{x - 1}{2}$.
The range of a function is the set of all possible output values (y-values).
The range of a function is the set of all possible output values (y-values).
The domain and range of a function are always the same.
The domain and range of a function are always the same.
The trigonometric function tangent (tan) is defined as $\frac{\text{opposite}}{\text{adjacent}}$ in a right triangle.
The trigonometric function tangent (tan) is defined as $\frac{\text{opposite}}{\text{adjacent}}$ in a right triangle.
The trigonometric function cosine (cos) is defined as $\frac{\text{opposite}}{\text{hypotenuse}}$ in a right triangle.
The trigonometric function cosine (cos) is defined as $\frac{\text{opposite}}{\text{hypotenuse}}$ in a right triangle.
The value of $\sin(\frac{\pi}{2})$ is equal to 1.
The value of $\sin(\frac{\pi}{2})$ is equal to 1.
Flashcards
Rational Number
Rational Number
A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
Irrational Number
Irrational Number
A number that cannot be expressed as a fraction p/q and has a non-repeating, non-terminating decimal representation.
Domain of a Function
Domain of a Function
Set of all possible input values (x-values) for which a function is defined.
One-to-One Function
One-to-One Function
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Inverse Function
Inverse Function
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Composite Function
Composite Function
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What is the value?
What is the value?
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Study Notes
Question 1
Part A: Classifying Numbers
- Rational numbers in the set are 2, 3 + 2.4, (√5)⁴, and 3.279...
- Irrational numbers in the set are 3√7, √2, and sin(π/4).
Part B: Solving Inequalities
- For 4 + 5(1 - 3x) ≥ 2 - 4x, the solution in interval notation is (-∞, 7/11].
- For x/(x + 2) ≥ 5, the solution in interval notation is (-5/2, -2).
- For (x² - 5)/(x² - x - 6) + 4/(5(x + 2)) ≥ 4/(5(x - 3)), the solution in interval notation is (-∞, -3)∪(-2, ∞).
- For √(6x - 5) ≤ x, the solution in interval notation is [1, ∞).
Question 2
Part A: Finding the Domain of Functions
- For f(x) = (3/x) - (5/(x+1)), the domain is (-∞, -1) ∪ (-1, 0) ∪ (0, ∞).
- For f(x) = ⁴√(x+1)/(1-x), the domain is [-1, 1).
- For f(x) = √(2x-6)/(x-5), the domain is [3, 5) ∪ (5, ∞).
- For f(x) = sin(x)/(x³-x), the domain is (-∞, -1) ∪ (-1, 0) ∪ (0, 1) ∪ (1, ∞).
Part B: Composition of Functions
- Given f(x) = √(9 - x) and g(x) = √x, the rule of f ∘ g is f(g(x)) = √(9 - √x).
- The domain of f ∘ g is [0, 81].
Question 3
Part A: Determining if Functions are the Same
- f(x) = √x²/(x+3) and g(x) = |x|/√(x+3) are not the same functions.
Part B: Determining if Functions are One-to-One
- f(x) = 3√1 - x⁵ is one-to-one.
- f(x) = (1 + 5x) / (4 - 7x) is one-to-one.
- f(x) = sin(x) is not one-to-one.
Part C: Finding the Inverse of a Function
- Given f(x) = (2x + 3) / (4x - 1), the inverse function f⁻¹(x) = (x + 3) / (4x - 2).
- The domain of f⁻¹ is R - {1/2}.
- The range of f⁻¹ is R - {1/4}.
Part D: Function Composition
- The function g(x) = 3√((2x + 3) / (4x - 1)) can be expressed as a composition of two functions: f(x) = 3√x and h(x) = (2x + 3) / (4x - 1).
Question 4
Part A: Finding Trigonometric Function Values
- Given tan x = -2 and sec x = √5, the remaining trigonometric functions can be found considering that x is in quadrant IV.
Part B: Verifying a Trigonometric Identity
- To verify that tan θ + cos θ / (1 + sin θ) = sec θ, manipulate the left-hand side to show it is equal to the right-hand side.
Part C: Finding Exact Values Without a Calculator
- The exact value of sin⁻¹(sin(5π/6)) is π/6
- The exact value of cos(π/4 + cos⁻¹(1/3)) is (√2 - 4√3)/6
- The exact value of sin(sin⁻¹(2/3) + 2sin⁻¹(-1/7)) is (94 - 2√6)/147
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