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Questions and Answers
If $n$ and $m$ are integers, and $a$ and $b$ are real numbers, which of the following statements is incorrect?
If $n$ and $m$ are integers, and $a$ and $b$ are real numbers, which of the following statements is incorrect?
- $a^n \cdot b^{-m} = a^n \div b^m$
- $a^{nm} = (a^m)^n$
- $(ab)^{nm} = a^{nm} \cdot b^{nm}$
- $a^n \cdot a^{-m} = a^{n-m}$ (correct)
- $a^n \cdot a^m = a^{n+m}$
The expression $\frac{1 - x - 12x^2}{1 - 9x^2}$ can be simplified to which of the following choices?
The expression $\frac{1 - x - 12x^2}{1 - 9x^2}$ can be simplified to which of the following choices?
- $\frac{1 - 3x}{1 + 2x}$
- $\frac{1 - x}{1 + 6x}$
- $\frac{4x - 1}{2x - 1}$
- $\frac{1 - 4x}{1 - 3x}$ (correct)
- $\frac{1 - 4x}{1 + 3x}$
$\frac{a+2}{2a-6} - \frac{a-2}{2a+6} =$ which of the following choices?
$\frac{a+2}{2a-6} - \frac{a-2}{2a+6} =$ which of the following choices?
- $\frac{9a}{2(a^2-9)}$
- $\frac{a+2}{4a}$
- $\frac{5a}{a^2-9}$ (correct)
- $\frac{1}{a}$
- $\frac{a^2+6}{a^2-9}$
Which of the following is a correct factoring of the expression $x^4 + 24x^2y^2 - 25y^4$ ?
Which of the following is a correct factoring of the expression $x^4 + 24x^2y^2 - 25y^4$ ?
If $y = 4x^2 – 5x + 4$, what is the value of $y$ when $x = 2$?
If $y = 4x^2 – 5x + 4$, what is the value of $y$ when $x = 2$?
If $y = \frac{x-2}{(x-3)(x+4)}$, $y$ cannot be evaluated for what value of $x$?
If $y = \frac{x-2}{(x-3)(x+4)}$, $y$ cannot be evaluated for what value of $x$?
What are the values of $x$ for which the following is true: $(x + 2)(x^2 − 1) = 0$?
What are the values of $x$ for which the following is true: $(x + 2)(x^2 − 1) = 0$?
The solution of the equation $\frac{3x-2}{5} = 4 - \frac{x}{2}$ is which of the following?
The solution of the equation $\frac{3x-2}{5} = 4 - \frac{x}{2}$ is which of the following?
One number is 5 more than another, and the sum of the two numbers is 25. What is the smaller of the two numbers?
One number is 5 more than another, and the sum of the two numbers is 25. What is the smaller of the two numbers?
Let $w = -2x + 4$. For what value or values of $x$ is $w > 0$?
Let $w = -2x + 4$. For what value or values of $x$ is $w > 0$?
Suppose $h(x) = x + 1$ and $g(h) = h^2 - 3$. What is $g$ as a function of $x$?
Suppose $h(x) = x + 1$ and $g(h) = h^2 - 3$. What is $g$ as a function of $x$?
Suppose $f(x) = x^2$ and $h(x) = x$. Which of the following is not true?
Suppose $f(x) = x^2$ and $h(x) = x$. Which of the following is not true?
Suppose $f(x) = 2x$ and $h(x) = 1$. For what $x$ value is it true that $f(x) - h(x) = 0$?
Suppose $f(x) = 2x$ and $h(x) = 1$. For what $x$ value is it true that $f(x) - h(x) = 0$?
Suppose $f(x) = x^3$ and $h(x) = x$. For what $x$ values is it true that $f(x) = h(x)$?
Suppose $f(x) = x^3$ and $h(x) = x$. For what $x$ values is it true that $f(x) = h(x)$?
Consider the function $f(x) = x^2 - 4$. For what $x$ is it true that $f(x) \ge 0$?
Consider the function $f(x) = x^2 - 4$. For what $x$ is it true that $f(x) \ge 0$?
Let $f(x) = \frac{x-2}{(x-3)(x+4)}$; $f(x)$ is not defined for which of the following values of $x$?
Let $f(x) = \frac{x-2}{(x-3)(x+4)}$; $f(x)$ is not defined for which of the following values of $x$?
Suppose $f(x) = (x +2)(x^2 – 1)$. For which of the following x values is it true that $f(x) < 0$?
Suppose $f(x) = (x +2)(x^2 – 1)$. For which of the following x values is it true that $f(x) < 0$?
Suppose $f(x) = \frac{3x-2}{x}$ and $g(x) = 4 - \frac{5}{2}$. For what $x$ value do these functions map into the same value?
Suppose $f(x) = \frac{3x-2}{x}$ and $g(x) = 4 - \frac{5}{2}$. For what $x$ value do these functions map into the same value?
A rectangle has a height $h$, width $w$, and a perimeter of length 30. Express $h$ as a function of $w$.
A rectangle has a height $h$, width $w$, and a perimeter of length 30. Express $h$ as a function of $w$.
Flashcards
Incorrect exponent rule?
Incorrect exponent rule?
Statement that is not always true for integers n and m, and real numbers a and b. The incorrect statement is anam = an+m
Simplifying Rational Expressions
Simplifying Rational Expressions
To simplify rational expressions, factor both numerator and denominator, then cancel out common factors.
When is a function not defined?
When is a function not defined?
A function y = f(x) cannot be evaluated where the denominator is zero.
What is Sine (sin)?
What is Sine (sin)?
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When is tangent undefined?
When is tangent undefined?
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What is cosine?
What is cosine?
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What is the fundamental Pythagorean identity?
What is the fundamental Pythagorean identity?
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Cosine function and periodicity
Cosine function and periodicity
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Study Notes
- The document is a readiness test for HSLDA Online Academy's AP® Calculus AB course.
- A Pre-Calculus and Trigonometry course is recommended if students require improvement.
- All courses have live, weekly class sessions taught by qualified instructors.
- The test assesses basic algebraic, function, and trigonometric skills.
- Students should work independently without a calculator and aim to answer at least 80% of the questions correctly to be considered ready for AP® Calculus AB.
Problems 1-10: Basic Algebraic Skills
- Question 1: Identifying the incorrect statement regarding integer exponents and real numbers.
- Question 2: Simplifying the expression (1 - x - 12x²) / (1 - 9x²).
- Question 3: Simplifying the expression (a+2)/(2a-6) = (a-2)/(2a+6).
- Question 4: Factoring the expression x⁴ + 24x²y² – 25y⁴.
- Question 5: Finding the value of y in the equation y = 4x² – 5x + 4 when x = 2.
- Question 6: Determining the x value for which y = (x-2)/((x-3)(x+4)) cannot be evaluated.
- Question 7: Finding the values of x for which (x + 2)(x² − 1) = 0 is true.
- Question 8: Solving the equation (3x-2)/5 = 4 - x/2.
- Question 9: Determining the smaller of two numbers given that one is 5 more than the other, and their sum is 25.
- Question 10: Finding the value(s) of x for which w = -2x + 4 > 0.
Problems 11-20: Basic Skills Involving Functions
- Question 11: Finding g as a function of x, given h(x) = x + 1 and g(h) = h² - 3.
- Question 12: Identifying the statement that is not true, given f(x) = x² and h(x) = x.
- Question 13: Identifying the statement that is not true, given f(x) = x⁻¹ and h(x) = x.
- Question 14: Finding the x value that satisfies f(x) - h(x) = 0, given f(x) = 2x and h(x) = 1.
- Question 15: Finding the x values for which f(x) = h(x) is true, given f(x) = x³ and h(x) = x.
- Question 16: Determining for which x values the function f(x) = x² - 4 ≥ 0.
- Question 17: Identifying for which x values the function f(x) = (x-2)/((x-3)(x+4)) is not defined.
- Question 18: Finding the x values for which f(x) = (x + 2)(x² - 1) < 0.
- Question 19: Finding the x value for which f(x) = (3x-2)/5 and g(x) = 4 - x/2 map into the same value.
- Question 20: Expressing the height h of a rectangle as a function of its width w, given its perimeter is 30.
Problems 21-30: Basic Skills Involving Trigonometry
- Questions 21-23 refer to triangle ABC with angles a, b, c; sides AB, BC, AC; and altitude BD to side AC.
- The length of a line is indicated with a pair of vertical lines |AB|.
- Question 21: Determining which of the following corresponds to sin(a).
- Question 22: Determining which of the following corresponds to cos(c).
- Question 23: Determining which of the following corresponds to tan(a).
- Question 24: Determining for which angle θ is tan(θ) not defined.
- Question 25: Given cos(θ) = sin(45°), determine for which θ value this is true.
- Question 26: Given tan(45°) = 1, determine for which θ value tan(θ) = 1 is also true.
- Question 27: Finding the minimum value of cos(θ + 45°).
- Question 28: Identifying which of the trigonometric statements is not true.
- Question 29: Given f(x) = cos(x) and n is an integer, determine which statement is true.
- Question 30: Considering the function cos(3θ), determine which θ value makes this function equal to 0.
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