Finite Math Examples

$\frac{2 x^{2} - 6}{{(x - 1)}^{2}} = f (x)$

Step 1

Find where the expression $y$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

Step 5

The oblique asymptote is a slant asymptote along the linear line of the graph, which is the linear portion of the long division result that was calculated when the expression was reduced.

$y = y$

Step 6

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = y$

Step 7