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.