A graphing device is recommended. Graph the rational function $$ f $$. $$ f(x)=\frac{2 x^{2}+10 x+10}{x+5}, g(x)=2 x $$ Determine all vertical asymptotes from your graph. (Enter your answers as a comma-separated list of equations.)

Question

UpStudy AI · Accepted Answer

**Step 1. Identify the function and potential vertical asymptotes**

We are given  
$$
f(x)=\frac{2x^{2}+10x+10}{x+5}.
$$

Vertical asymptotes occur where the denominator is zero (provided the zero does not cancel with a corresponding factor in the numerator).

**Step 2. Find the zeros of the denominator**

Set the denominator equal to zero:
$$
x+5=0.
$$
Solving for $$ x $$:
$$
x=-5.
$$

**Step 3. Check for common factors**

We need to check if the factor $$ x+5 $$ cancels with something in the numerator. Factor the numerator or check if $$ x=-5 $$ is a zero of the numerator:
$$
2(-5)^2 + 10(-5) + 10 = 2(25) - 50 + 10 = 50 - 50 + 10 = 10.
$$
Since the numerator does not vanish at $$ x=-5 $$, there is no cancellation.

**Step 4. Conclusion**

Thus, the only vertical asymptote is given by
$$
x=-5.
$$

**Final Answer:**  
$$x=-5$$
The vertical asymptote is $$ x = -5 $$.

A graphing device is recommended. Graph the rational function \( f \). \[ f(x)=\frac{2 x^{2}+10 x+10}{x+5}, g(x)=2 x \] Determine all vertical asymptotes from your graph. (Enter your answers as a comma-separated list of equations.)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions