Determine the vertical asymptote(s) of the function . If there is more
than one vertical asymptote, seperate your answers with a comma.

Ask by Savage Hart. in the United States

Nov 22,2024

Question

Determine the vertical asymptote(s) of the function $$ f(x)=\frac{4 x}{x^{2}+6 x-7} $$. If there is more than one vertical asymptote, seperate your answers with a comma.

UpStudy AI · Accepted Answer

To determine the vertical asymptotes of the function $$ f(x) = \frac{4x}{x^2 + 6x - 7} $$, we need to find the values of $$ x $$ that make the denominator equal to zero, as vertical asymptotes occur where the function is undefined due to division by zero.

First, we set the denominator equal to zero:

$$
x^2 + 6x - 7 = 0
$$

Next, we can factor the quadratic equation. We look for two numbers that multiply to $$-7$$ (the constant term) and add to $$6$$ (the coefficient of $$x$$). The numbers $$7$$ and $$-1$$ satisfy these conditions:

$$
x^2 + 6x - 7 = (x + 7)(x - 1)
$$

Now, we set each factor equal to zero to find the values of $$x$$:

1. $$x + 7 = 0 \Rightarrow x = -7$$
2. $$x - 1 = 0 \Rightarrow x = 1$$

Thus, the vertical asymptotes of the function $$ f(x) $$ are at:

$$
x = -7, \quad x = 1
$$

Therefore, the vertical asymptotes are:

$$
\boxed{-7, 1}
$$
The vertical asymptotes are at $$ x = -7 $$ and $$ x = 1 $$.

Determine the vertical asymptote(s) of the function . If there is more
than one vertical asymptote, seperate your answers with a comma.

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Determine the vertical asymptote(s) of the function . If there is more than one vertical asymptote, seperate your answers with a comma.