4.5 – Rational Functions and Inequalities

Rational Function = a function which may be written in the form, where p(x) and q(x) are both polynomial functions The domain for any rational function is the set of all x such that q(x) ≠ 0

With rational functions, we may have occurrences of so-called asymptotes – An additional line/reference point which the graph of the function will approach With all asymptotes, they will approach the line, but never touch the line nor cross/intersect it

Vertical Asymptote The vertical line x = c is a vertical asymptote if f(x) increases in magnitude without bound (takes off infinitely) as x approaches c

Horizontal Asymptote The horizontal line y = c is a horizontal asymptote if f(x) approaches the value as x -> -∞ or as x -> ∞

Oblique Asymptote A non-vertical OR non-horizontal line may also be an asymptote; also known as slant asymptote

We can determine the location of asymptotes by investigating the ratio of – p is an nth degree polynomial – q is an mth degree polynomial – a n = leading coefficient of p(x) – b m = leading coefficient of q(x)

To find a Horizontal Asymptote: if n < m (degree of top less than the bottom degree), the line y = 0 is an horizontal asymptote If n = m, then y = (a n /b m ) is the horizontal asymptote

Example. Determine the horizontal asymptotes for: A) f(x) = B) g(x) = C) h(x) =

To find a vertical asymptote: if has been reduced (no more common factors), then the line x = c is a vertical asymptote if c is a real zero of q(x) Find the zeros (factored form) for q(x); these are the lines for your vertical asymptotes

Example. Determine the vertical asymptotes for: A) B)

Oblique If n = m + 1, from the rational function : The line y = g(x) is the oblique asymptote, where g(x) is the polynomial quotient – Must divide the two polynomials – Remainder doesn’t matter

Example. Determine the oblique asymptote for the function

Assignment Pg odd, odd