Sets of Numbers and Inequalities

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Questions and Answers

How does the presence of absolute value in an inequality, such as $|2x + 4| \leq 3$, affect the solution process compared to a linear inequality without absolute value?

Absolute value inequalities require considering two cases: one where the expression inside the absolute value is positive or zero, and another where it is negative, leading to two separate inequalities to solve.

In the context of solving inequalities, explain how the sign of the coefficient of $x$ affects the direction of the inequality when multiplying or dividing both sides by a negative number, using the example $4 - 5x < 7$.

When multiplying or dividing by a negative number, the direction of the inequality sign must be reversed. In this case, dividing by -5 would require flipping the '<' sign.

Describe the difference in the solution set between a compound inequality joined by 'and' versus one joined by 'or', using an example like $-3 < 2x + 1 \leq 7$.

An 'and' compound inequality requires both inequalities to be true simultaneously, resulting in an intersection of solution sets. An 'or' requires at least one to be true giving a union of solution sets.

Explain the potential impact of a discontinuity on the domain of a rational function, and how this relates to finding the domain of a function like $f(x) = \frac{2x}{x^2 - 4}$?

Discontinuities, such as those caused by division by zero in rational functions, exclude certain x-values from the domain. In this example, $x = 2$ and $x = -2$ would be excluded. Signup and view all the answers

How does the presence of a square root in a function, such as $f(x) = \sqrt{3x - 9}$, restrict the domain of the function, and what algebraic step is necessary to determine this restriction?

The expression inside the square root must be non-negative. Setting $3x - 9 \geq 0$ and solving for <code>$x$</code> gives the domain restriction. Signup and view all the answers

Describe how to determine if a function is even, odd or neither. Show your work and thinking.

A function is even if $f(-x) = f(x)$, odd if $f(-x) = -f(x)$, and neither if neither of these conditions hold. Replace x with -x and simplify f(x) to see if it meets the conditions. Signup and view all the answers

How does knowledge of the unit circle assist in evaluating trigonometric functions, specifically when finding the exact value of $\sin(\frac{17\pi}{6})$?

The unit circle provides the coordinates of points on the circle corresponding to specific angles, which directly relate to sine and cosine values. Find the reference angle to determine the value of the trig function. Signup and view all the answers

What is the purpose of the horizontal line test in determining whether a function has an inverse. What are the implications if the function fails the test?

The horizontal line test checks if any horizontal line intersects the graph of the function more than once. If it does the functions is not one to one and does not have an inverse. Signup and view all the answers

Explain how finding a common denominator is essential when evaluating limits of combined rational function. Give an example.

A common denominator allows the combination of fractions into a single expression. For example, $\lim_{x \to 0} \frac{1}{x} + \frac{2}{x+1}$ becomes $\lim_{x \to 0} \frac{x+1 + 2x}{x(x+1)} = \frac{3x+1}{x(x+1)}$ Signup and view all the answers

Describe, conceptually, what the Squeeze Theorem allows you to conclude about the limit of a function $h(x)$ at a point $c$, given that $g(x) \leq h(x) \leq f(x)$ near $c$, and $\lim_{x \to c} g(x) = \lim_{x \to c} f(x) = L$.

The Squeeze Theorem states that if $h(x)$ is bounded between two functions, $f(x)$ and $g(x)$, that both approach the same limit $L$ as $x$ approaches $c$, then $h(x)$ must also approach the limit $L$ at $c$. Signup and view all the answers

In the context of limits involving infinity, explain the strategy used to evaluate the $\lim_{x \to \infty} \frac{3x + 1}{x - 5}$, and how it simplifies the expression to find the limit.

Divide both the numerator and the denominator by the highest power of $x$ in the denominator. In the example, dividing by $x$ results in $\lim_{x \to \infty} \frac{3 + \frac{1}{x}}{1 - \frac{5}{x}}$, which simplifies to 3 as $x$ approaches infinity. Signup and view all the answers

Describe the relationship between limits and continuity. What condition involving limits must be satisfied for a function, f(x), to be continuous at x=a?

For f(x) to be continuous at x=a 1) f(a) must exist; 2) the $\lim_{x \to a} f(x)$ exist; 3) $\lim_{x \to a} f(x) = f(a)$. Signup and view all the answers

What does it mean for a function to be one-to-one? Explain in terms of inputs, outputs and the graph of the function.

A function is one-to-one if each input has a unique output. Graphically, it means the function passes the Horizontal Line Test. Signup and view all the answers

Describe two different scenarios where you might need to use the Squeeze Theorem to find a limit.

When dealing with functions that oscillate rapidly, or when bounding a complicated function with simpler functions whose limits are known Signup and view all the answers

What exactly is a reference angle and how does it simplify trigonometric calculations.

The reference angle ia acute version of any angle, formed by the terminal side of the angle and the x-axis. Trig values become easy to calculate using knowledge of first quadrant values. Signup and view all the answers

If the $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = M$ where $L \neq M$, what can you conclude about $\lim_{x \to a} f(x)$?

$\lim_{x \to a} f(x)$ does not exist. Signup and view all the answers

Why is it essential to consider both left-hand and right-hand limits when evaluating the limit of a function at a point, and what does it imply if they are not equal?

Considering both sides ensures the function approaches the same value from either direction. If they aren't equal, the limit doesn't exist. Signup and view all the answers

What is the domain of a function and how does its determination differ for polynomial, rational, and radical functions?

The domain is the set of all possible input values. Polynomial functions are all reals; rational functions exclude values that make the denominator zero; radical functions require the radicand (expression inside) to be non-negative. Signup and view all the answers

How does using the properties of inverse trig functions simplify evaluating an expression such as cos(cos$^{-1}$($\pi$/4))

Since the value $\pi$/4 doesn't fall between -1 and 1, cos(cos$^{-1}$($\pi$/4)) doesn't yield the identity. It's outside the defined domain and so this problem has no solution. Signup and view all the answers

What is the importance of rationalizing the numerator or denominator when evaluating certain limits. Provide and example.

Rationalizing can help remove indeterminate forms like $\frac{0}{0}$ by algebraically manipulating expression. For example, $\lim_{x \to 0} \frac{\sqrt{x+9}-3}{x}$ Signup and view all the answers

Flashcards

What is an inequality?

An inequality is a statement that compares two expressions using inequality symbols.

How to solve an inequality?

To solve an inequality, find the values of the variable that make the inequality true.