Questions: For the function f(x)=(x-4)/((-2 x+6)(4 x+10)) What are the vertical asymptotes? Give a list of the x-values of the asymptotes separated by commas. x= What is the horizontal asymptote? y=

For the function
f(x)=(x-4)/((-2 x+6)(4 x+10))

What are the vertical asymptotes? Give a list of the x-values of the asymptotes separated by commas.
x=

What is the horizontal asymptote?
y=

Transcript text: For the function \[ f(x)=\frac{x-4}{(-2 x+6)(4 x+10)} \] What are the vertical asymptotes? Give a list of the $x$-values of the asymptotes separated by commas. \[ x=\square \] What is the horizontal asymptote? \[ y=\square \]

Solution

Solution Steps

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is equal to zero, provided the numerator is not zero at those points. For the function

\[ f(x) = \frac{x-4}{(-2x+6)(4x+10)} \]

we need to find the values of $x$ that make the denominator zero:

\[ (-2x + 6)(4x + 10) = 0 \]

This gives us two equations to solve:

$-2x + 6 = 0$
$4x + 10 = 0$

Solving these:

$-2x + 6 = 0 \Rightarrow x = 3$
$4x + 10 = 0 \Rightarrow x = -\frac{5}{2}$

Thus, the vertical asymptotes are at $x = 3$ and $x = -\frac{5}{2}$.

Step 2: Identify Horizontal Asymptote

The horizontal asymptote of a rational function is determined by the degrees of the polynomials in the numerator and the denominator. The function is:

\[ f(x) = \frac{x-4}{(-2x+6)(4x+10)} \]

The degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.