Rational Inequalities and Equations Quiz

Rational Inequality

• A rational inequality is an inequality that contains a rational expression.
• To solve a rational inequality, first rewrite the inequality in the form of a single rational expression set to zero or less than (or greater than) zero.
• Identify the critical values (zeros of the numerator and denominator). These are the values that make the rational expression undefined or equal to zero.
• Create a sign chart using the critical values to determine the intervals where the expression is positive or negative.
• Test a value from each interval in the original inequality to confirm the solution set.

Rational Equations

• A rational equation is an equation that contains a rational expression. in
• To solve a rational equation, first find the least common denominator (LCD) of all the rational expressions in the equation.
• Multiply both sides of the equation by the LCD to eliminate the fractions.
• Solve the resulting polynomial equation.
• Check your solutions to ensure that they do not make any denominator equal to zero.

Rational Function (Video)

• A rational function is a function that can be expressed as the quotient of two polynomial functions.
• The domain of a rational function excludes any values that make the denominator equal to zero.
• Rational functions can have asymptotes (vertical, horizontal, or slant asymptotes). The locations of these asymptotes are often useful for graphing the function.
• A rational function can also have holes in its graph which are removable discontinuities in its points but they aren't asymptotes since the points won't be undefined.

Table of Values

• Create a table of values to help graph a function.
• Pick representative x-values across the domain of the function.
• Substitute each x-value into the function to find the corresponding y-value.
• Record both x- and y-values in the table.
• Plot the points on a coordinate plane.
• Connect the points to form a graph of the function.

Inverse Function (Video)

• The inverse function "undoes" the original function.
• The domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse function.
• To find an inverse function, swap x and y in the function's equation, then solve for y.
• Verify the result by checking whether (f(x)) and (f⁻¹(x)) satisfy the condition (f⁻¹(f(x)) = x) and (f(f⁻¹(x)) = x).
• Graphing the inverse function is accomplished by reflecting the graph of the original function across the line y=x.

Test your knowledge on rational inequalities and equations. This quiz covers the steps to solve these types of mathematical expressions, including rewriting inequalities, identifying critical values, and using least common denominators. Assess your understanding through a series of challenging questions.