Exploring Rational Functions and Their Operations

What is a rational function?

• A function that cannot have any asymptotes.

• A function that always crosses the x-axis.

• A function that can only be expressed using rational numbers.

• A function represented by a polynomial in both numerator and denominator.

What is a rational number?

• Any number that can be expressed as a fraction.

• A number that cannot be expressed as a fraction.

• A number that includes pi and e.

• A number that can be expressed as an integer over another integer.

What causes a vertical asymptote in a rational function?

• Values that make the numerator zero.

• Values that make the denominator zero.

• Values where the function intersects the y-axis.

• Values that result in undefined horizontal asymptotes.

What is the domain of a rational function?

• Only positive real numbers.

• All real numbers except where the denominator equals zero.

• All real numbers except where the function intersects the y-axis.

• All real numbers.

What happens to the graph of a rational function at a vertical asymptote?

• The graph approaches the asymptote but never touches or crosses it.

• The graph touches the asymptote but does not cross it.

• The graph becomes undefined.

• The graph crosses the asymptote.

How do you find a horizontal asymptote when the degrees of the numerator and denominator are the same?

• By subtracting the leading coefficient of the denominator from the numerator.

• By adding the degrees of the numerator and denominator.

• By dividing the leading coefficients of the numerator by the denominator.

• Horizontal asymptotes do not exist in this case.

When the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is:

• Non-existent

• y = 0

• Dependent on the leading coefficients

• y = 1

What is the significance of the leading coefficients in determining horizontal asymptotes?

• They have no significance in determining asymptotes.

• They are divided to find the horizontal asymptote when degrees of numerator and denominator are equal.

• They are used to calculate the y-intercept of the asymptote.

• They determine the slope of the asymptote.

Under what condition does a slant asymptote occur?

• Slant asymptotes occur regardless of the degrees of the numerator and denominator.

• When the degree of the numerator is less than the degree of the denominator.

• When the degree of the numerator is greater than the degree of the denominator by more than one.

• When the degree of the numerator is exactly one more than the degree of the denominator.

Which of the following is true about slant asymptotes?

• They are always horizontal lines.

• They occur when the numerator's degree is less than the denominator's degree.

• They can occur when the numerator's degree is one more than the denominator's degree.

• They are determined by setting the numerator equal to zero.