Question

hw 9 rational functions
due wednesday by 11:59pm points 100 submitting an external tool
which functions have graphs with a slant asymptote? select all that apply.
select all that apply:
$f(x)=\frac{7x^{2}}{x^{3}-5}$
$f(x)=\frac{2x^{2}+1}{3x^{2}}$
$f(x)=\frac{2x^{5}+1}{x^{4}-2}$
$f(x)=\frac{x^{2}+1}{4 - x}$
$f(x)=\frac{x^{4}+2}{x^{5}-3}$

Explanation:

Step1: Recall slant - asymptote condition

A rational function $y = \frac{f(x)}{g(x)}$ has a slant asymptote when the degree of the numerator $f(x)$ is exactly one more than the degree of the denominator $g(x)$.

Step2: Analyze $f(x)=\frac{7x^{2}}{x^{3}-5}$

Degree of numerator is 2, degree of denominator is 3. Since $2<3$, no slant asymptote.

Step3: Analyze $f(x)=\frac{2x^{2}+1}{3x^{2}}$

Degree of numerator is 2, degree of denominator is 2. Since $2 = 2$, no slant asymptote.

Step4: Analyze $f(x)=\frac{2x^{5}+1}{x^{4}-2}$

Degree of numerator is 5, degree of denominator is 4. Since $5=4 + 1$, has a slant asymptote.

Step5: Analyze $f(x)=\frac{x^{2}+1}{4 - x}$

Degree of numerator is 2, degree of denominator is 1. Since $2=1 + 1$, has a slant asymptote.

Step6: Analyze $f(x)=\frac{x^{4}+2}{x^{5}-3}$

Degree of numerator is 4, degree of denominator is 5. Since $4<5$, no slant asymptote.

Answer:

$f(x)=\frac{2x^{5}+1}{x^{4}-2}$, $f(x)=\frac{x^{2}+1}{4 - x}$