{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}


Though not absolutely necessary4 it is good practice

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f Vertical Asymptotes and Holes:a Suppose r is a rational function ( x) which can be written as r(x) = p(x) where p and q have no common zeros.b Let c be a real number q which is not in the domain of r. (c) • If q (c) = 0, then the graph of y = r(x) has a hole at c, p(c) . q • If q (c) = 0, then the the line x = c is a vertical asymptote of the graph of y = r(x). a b Or, ‘How to tell your asymptote from a hole in the graph.’ In other words, r(x) is in lowest terms. In English, Theorem 4.1 says if x = c is not in the domain of r but, when we simplify r(x), it no longer makes the denominator 0, then we have a hole at x = c. Otherwise, we have a vertical asymptote. Example 4.1.2. Find the vertical asymptotes of, and/or holes in, the graphs of the following rational functions. Verify your answers using a graphing calculator. 1. f (x) = 2x 2−3 x 3. h(x) = x2 − x − 6 x2 + 9 2. g (x) = x2 − x − 6 x2 − 9 4. r(x) = x2 − x − 6 x2 + 4 x + 4 Solution. 1. To use Theorem 4.1, we first find all of the √ numbers which aren’t in the domain of f . To real 2 − 3 = 0 and get x = ± 3. Since the expression f (x) is in lowest terms, do so, we solve x √ √ there is no...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online