# Give the range of the rational functions given that neither of the graphs crosses its horizontal asymptote. Write your answers in set notation and int

Give the range of the rational functions given that neither of the graphs crosses its horizontal asymptote. Write your answers in set notation and interval notation.

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Nathanael Webber

1. For a rational function, the range is all real numbers, except the y values that correspond to holes or horizontal asymptotes.
In Example B, part(b), the horizontal asymptote was $$y = 2$$. Therefore, the range is all real numbers, except $$y = 2$$
The set notation for the range is $$\displaystyle{\left\lbrace{y}\in{R}{\mid}{y}\ne{q}_{2}\right\rbrace}$$
The interval notation is $$\displaystyle{y}\in{\left(—\infty,{2}\right)}\cup{\left({2},\infty\right)}$$
2. In ‘Try These B, part(a), we were given
$$\displaystyle{y}={\frac{{{2}-{x}}}{{{x}+{4}}}}$$
In a rational function, if the degree of the numerator and the denominator are the same, then the horizontal asymptote is given by
$$y =$$ (ratio of the leading coefficients in the numerator/denominator)
In this case, we have
$$\displaystyle{y}={\frac{{-{1}}}{{{1}}}}=-{1}$$
Therefore, the set notation for the range is $${y ¢ R| y -1}$$
The interval notation is $$\displaystyle{y}\in{\left(—\infty,-{1}\right)}\cup{\left(-{1},\infty\right)}$$