10.(12 pts) Find a rational function $$ g(x) $$ such that all of the following hold: a. $$ g(x) $$ has vertical asymptotes at $$ x=2 $$ and $$ x=1 $$ b. $$ g(x) $$ has $$ x $$-intercepts at $$ x=-6 $$ and $$ x=-5 $$ c. $$ g(x) $$ has a $$ y $$-intercept at 6

Question

UpStudy AI · Accepted Answer

1. Write a general rational function with the given zeros and vertical asymptotes:
   
   $$
   g(x) = A\frac{(x+6)(x+5)}{(x-2)(x-1)}
   $$
   
   where $$ A $$ is a constant to be determined.

2. Use the given $$ y $$-intercept at 6. Since the $$ y $$-intercept corresponds to $$ g(0) = 6 $$, substitute $$ x = 0 $$:

$$
   g(0) = A\frac{(0+6)(0+5)}{(0-2)(0-1)} = A\frac{6 \cdot 5}{(-2)(-1)}.
   $$

3. Simplify the expression:

$$
   g(0) = A\frac{30}{2} = 15A.
   $$

4. Set the expression equal to 6 and solve for $$ A $$:

$$
   15A = 6 \quad \Longrightarrow \quad A = \frac{6}{15} = \frac{2}{5}.
   $$

5. Substitute $$ A = \frac{2}{5} $$ back into the function:

$$
   g(x) = \frac{2}{5}\frac{(x+6)(x+5)}{(x-2)(x-1)}.
   $$
$$ g(x) = \frac{2}{5} \cdot \frac{(x+6)(x+5)}{(x-2)(x-1)} $$

10.(12 pts) Find a rational function such that all of the following hold:
a. has vertical asymptotes at and
b. has -intercepts at and
c. has a -intercept at 6

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions

10.(12 pts) Find a rational function such that all of the following hold: a. has vertical asymptotes at and b. has -intercepts at and c. has a -intercept at 6