Question

Write a rational function $f$ that has vertical asymptote at $x=4$, a horizontal asymptote at $y=5$ and a zero at $x=-7$.

• A. $f\left(x\right)=\frac{5(x-7)}{(x-4)}$
• B. $f\left(x\right)=\frac{5(x+7)}{(x-4)}$
• C. $f\left(x\right)=\frac{5(x-7)}{(x+4)}$
• D. $f\left(x\right)=\frac{(x+7)}{(x+4)}$

Answer 1

The correct option is B $f\left(x\right)=\frac{5(x+7)}{(x-4)}$

Since the rational function $f$ has the vertical asymptote at $x=4$, then the denominator of $f$ contains the term $(x-4)$.

Thus function $f\left(x\right)$ is of the form $f=\frac{g\left(x\right)}{x-4}$.

Since the horizontal asymptote exists $y=5$, the numerator $g\left(x\right)$ of $f\left(x\right)$ has to be of the same degree as the denominator with a leading coefficient equal to $5$.

Also $g\left(x\right)$ must contain the term $(x+7)$ since $f$ has zero at $x=-7$.

Hence, $f\left(x\right)=\frac{5(x+7)}{(x-4)}$