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

sagnuhh 2021-06-27 Answered
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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Expert Answer

Nathanael Webber
Answered 2021-06-28 Author has 12471 answers

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 \(\displaystyle{y}\in{\left(—\infty,-{1}\right)}\cup{\left(-{1},\infty\right)}\)

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