03/10 Rational Functions Free Online FlashCards

What is a rational function?

A rational function has the form f(x)=p(x)/q(x), where p(x) and q(x) are polynomials and q(x)≠0.

How do you simplify a rational expression?

Factor both polynomials, record zeros of the original denominator, cancel common factors, and retain the original restrictions.

How is the domain of a rational function found?

The domain excludes every real number that makes the original denominator equal to zero.

What creates a hole in a rational-function graph?

A canceled denominator factor creates a removable discontinuity, represented graphically by a hole.

What produces a vertical asymptote?

A remaining zero of the denominator after simplification produces a vertical asymptote, where the function becomes unbounded near that input.

How do degrees determine a horizontal asymptote?

If the numerator degree is lower, y=0; if degrees are equal, y is the ratio of leading coefficients; if the numerator degree is higher, there is no horizontal asymptote.

What is the slant asymptote of (x²+1)/(x−1)?

For h(x)=(x²+1)/(x−1), division gives h(x)=x+1+2/(x−1), so the slant asymptote is y=x+1.

How is an x-intercept found for a rational function?

For f(x)=(x−2)/(x+1), the x-intercept is (2,0), found by setting the simplified numerator equal to zero.

When can you find a y-intercept?

The y-intercept is found by evaluating f(0), provided 0 belongs to the domain. For (x−2)/(x+1), it is (0,−2).

What is the safe method for solving rational equations?

Factor denominators, state restrictions, find the LCD, multiply every term by it, solve, and check candidates in the original equation.

What is an extraneous solution?

An extraneous solution is an algebraic candidate that is not allowed in the original equation, usually because it makes a denominator zero.

How do you solve a rational inequality?

Move all terms to one side, factor, identify numerator and denominator zeros, test each resulting interval, and apply endpoint rules.