Understanding Rational Functions

What is a rational function?

• A rational function is a function that involves irrational numbers in its expression.

• A rational function is a function that can only be graphed using polar coordinates.

• A rational function is a function that can be expressed as the quotient of two polynomial functions, where the denominator function is not equal to zero.

• A rational function is a function that has a linear equation as its output.

What is the general form of a rational function?

• f(x) = (ax^n + bx^(n-1) + ... + k) / (cx^m + dx^(m-1) + ... + l) + sin(x)

• f(x) = (ax^n + bx^(n-1) + ... + k) / (cx^m + dx^(m-1) + ... + l) + mx + b

• f(x) = (ax^n + bx^(n-1) + ... + k) / (cx^m + dx^(m-1) + ... + l)

• f(x) = (ax^n + bx^(n-1) + ... + k) / (cx^m + dx^(m-1) + ... + l) + 1/x

What is the domain of a rational function?

• All real numbers except for the values that make the denominator zero

• All integers

• Values greater than 10

• Only positive numbers

How do you find the vertical asymptotes of a rational function?

• Take the derivative of the rational function.

• Graph the rational function and look for vertical lines.

• Find the x-intercepts of the rational function.

• Set the denominator of the rational function equal to zero and solve for the variable.

How do you find the horizontal asymptotes of a rational function?

• Find the average of the numerator and denominator.

• Compare the degrees of the numerator and denominator of the rational function.

• Divide the numerator by the denominator.

• Count the number of x-intercepts.

What is a removable discontinuity in a rational function?

• A removable discontinuity in a rational function is when the function has a horizontal asymptote

• A removable discontinuity in a rational function is when the function has a vertical asymptote

• A removable discontinuity in a rational function is when there is a hole in the graph due to a factor in the denominator canceling out with a factor in the numerator.

• A removable discontinuity in a rational function is when the function is continuous everywhere

How do you simplify a rational function?

• Divide the numerator and denominator by a common factor

• Add the numerator and denominator together

• Factor both the numerator and denominator, cancel out common factors, simplify

• Multiply the numerator and denominator by a common factor

What is the relationship between the degree of the numerator and denominator in a rational function?

• The degree of the numerator is less than, equal to, or greater than the degree of the denominator.

• The degree of the numerator is always greater than the degree of the denominator

• The degree of the numerator is always equal to the degree of the denominator

• The degree of the numerator is always less than the degree of the denominator

How do you graph a rational function?

• Step 3: Identify vertical asymptotes

• Step 1: Identify vertical asymptotes Step 2: Find horizontal asymptote Step 3: Determine x-intercepts Step 4: Plot key points and sketch the graph

• Step 1: Determine horizontal asymptotes

• Step 2: Locate x-intercepts

What is the significance of the x-intercepts in a rational function?

• The x-intercepts in a rational function are imaginary numbers

• The x-intercepts in a rational function represent the points where the function crosses the x-axis, indicating the roots of the function.

• The x-intercepts in a rational function have no mathematical significance

• The x-intercepts in a rational function are always positive