Question

Find the domain of each function. Find any horizontal, vertical, or oblique asymptotes.

$g\left(x\right)=\frac{2x^2-14x+24}{x^2+6x-40}$

Short answer

The domain of the function is$\left\{x|x\ne -10,4\right\}$, vertical asymptote is$x=-10$and horizontal asymptote is$y=2$

Step-by-step solution

Step 1. Given Information

The given function is$g\left(x\right)=\frac{2x^2-14x+24}{x^2+6x-40}$

Step 2. Calculation

Substitute the denominator expression equal to 0 to find the domain.

$$\begin{gathered} x^2+6x-40=0 \\ (x-4)(x+10)=0 \\ x=4,-10 \end{gathered}$$

Thus, the domain of the function is$\left\{x|x\ne -10,4\right\}$

Step 3. Calculation

Factor the numerator and denominator to make the function in lowest terms.

$$\begin{gathered} g\left(x\right)=\frac{2x^2-14x+24}{x^2+6x-40} \\ =\frac{2(x-4)(x-3)}{(x-4)(x+10)} \\ =\frac{2(x-3)}{(x+10)} \end{gathered}$$

$(x+10)$is the only factor in denominator so, vertical asymptote is $x=-10$.

The horizontal asymptote can be found by dividing the leading coefficients of the numerator and denominator.

$\frac{2}{1}=2$

Thus, horizontal asymptote is$y=2$