Solving Polynomial and Rational Inequalities (and review)

What does it mean for a function to be continuous at a point?

• A function is continuous at a point x = c if f(c) is defined, the limit exists, and the limit equals f(c).

• A function is continuous at a point if it has a derivative at that point.

• A function is continuous if it is defined for all real numbers.

• A function is continuous if it oscillates between two values.

What are the types of discontinuities?

• Infinite Discontinuity: The function approaches infinity at a point.

• Continuous Discontinuity: The function is defined at all points.

• Jump Discontinuity: The function has a sudden change in value at a point.

• Oscillating Discontinuity: The function oscillates infinitely at a point.

What is a slant asymptote?

• A horizontal line that a graph approaches as x approaches infinity.

• A diagonal line that a graph approaches as x approaches infinity or negative infinity.

• A vertical line that a graph approaches as x approaches a certain value.

• A constant value that a graph approaches as x approaches infinity.

What is the significance of the roots in solving inequalities?

• The roots indicate the maximum value of the expression.

• The roots of the polynomial or rational expression divide the number line into intervals. The sign of the expression in each interval determines where the inequality is satisfied.

• The roots are irrelevant in solving inequalities.

• The roots provide the exact solutions to the inequality.

How do you express the solution set of an inequality?

• In interval notation, set-builder notation, or graphically on a number line.

• Only in interval notation.

• Only graphically on a number line.

• In a table format.

What is the importance of the leading coefficient in polynomial inequalities?

• It determines the degree of the polynomial.

• It affects the number of real roots of the polynomial.

• It determines the end behavior of the polynomial. If it is positive, the polynomial approaches positive infinity as x approaches positive or negative infinity; if negative, it approaches negative infinity.

• It indicates whether the polynomial is even or odd.

What is the graphical representation of polynomial inequalities?

• It shows the regions where the polynomial is above or below the x-axis, indicating where the inequality holds true.

• It represents the roots of the polynomial only.

• It displays the polynomial's coefficients in a bar graph format.

• It illustrates the polynomial's behavior at infinity.

What is interval notation?

• Interval notation is a way of writing subsets of the real number line, where (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes a and b.

• Interval notation is a method for solving equations involving variables.

• Interval notation is a technique used to graph linear functions.

• Interval notation is a way to express complex numbers.

What is a test point in the context of inequalities?

• A value chosen from an interval created by the roots of the inequality, used to determine whether the inequality holds true in that interval.

• A point that represents the maximum value of the inequality.

• A point that indicates where the inequality changes direction.

• A value that is always greater than the roots of the inequality.

How do you find the slant asymptote of a rational function?

• By performing polynomial long division and using the quotient (ignoring the remainder) to find the equation of the slant asymptote.

• By finding the roots of the denominator and setting them equal to zero.

• By taking the derivative of the function and analyzing its behavior at infinity.

• By evaluating the function at various points and plotting the results.