Question

follow the seven step strategy to graph the following rational function.
$f(x) = \frac{3x}{x^2 - 4}$

find the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to compl choice.
a. the equation(s) of the horizontal asymptote(s) is/are $y = 0$.
(type an equation. use a comma to separate answers as needed.)
b. there is no horizontal asymptote.

plot points between and beyond each x-intercept and vertical asymptote. find the value of the function at the x.
$x$: -7, -6, -1, 1, 5, 6
$f(x) = \frac{3x}{x^2 - 4}$: $\square$, $\square$, $\square$, $\square$, $\square$, $\square$
(simplify your answers.)

Explanation:

Step1: Substitute x = -7

Substitute \( x = -7 \) into \( f(x) = \frac{3x}{x^2 - 4} \).
\( f(-7) = \frac{3(-7)}{(-7)^2 - 4} = \frac{-21}{49 - 4} = \frac{-21}{45} = -\frac{7}{15} \)

Step2: Substitute x = -6

Substitute \( x = -6 \) into \( f(x) = \frac{3x}{x^2 - 4} \).
\( f(-6) = \frac{3(-6)}{(-6)^2 - 4} = \frac{-18}{36 - 4} = \frac{-18}{32} = -\frac{9}{16} \)

Step3: Substitute x = -1

Substitute \( x = -1 \) into \( f(x) = \frac{3x}{x^2 - 4} \).
\( f(-1) = \frac{3(-1)}{(-1)^2 - 4} = \frac{-3}{1 - 4} = \frac{-3}{-3} = 1 \)

Step4: Substitute x = 1

Substitute \( x = 1 \) into \( f(x) = \frac{3x}{x^2 - 4} \).
\( f(1) = \frac{3(1)}{(1)^2 - 4} = \frac{3}{1 - 4} = \frac{3}{-3} = -1 \)

Step5: Substitute x = 5

Substitute \( x = 5 \) into \( f(x) = \frac{3x}{x^2 - 4} \).
\( f(5) = \frac{3(5)}{(5)^2 - 4} = \frac{15}{25 - 4} = \frac{15}{21} = \frac{5}{7} \)

Step6: Substitute x = 6

Substitute \( x = 6 \) into \( f(x) = \frac{3x}{x^2 - 4} \).
\( f(6) = \frac{3(6)}{(6)^2 - 4} = \frac{18}{36 - 4} = \frac{18}{32} = \frac{9}{16} \)

Answer:

For \( x = -7 \): \( -\frac{7}{15} \)
For \( x = -6 \): \( -\frac{9}{16} \)
For \( x = -1 \): \( 1 \)
For \( x = 1 \): \( -1 \)
For \( x = 5 \): \( \frac{5}{7} \)
For \( x = 6 \): \( \frac{9}{16} \)