Questions: Plot points between and beyond each (x)-intercept and vertical asymptote. Find the value of the function at the given value of (x). (x: -4, -3, -1/2, 1/2, 3, 4) (f(x) = 6x^2 / (x^2 - 1)) (Simplify your answers.)

Solution Steps

Given the function \( f(x) = \frac{6x^2}{x^2 - 1} \), we substitute \( x = -4 \): \[ f(-4) = \frac{6(-4)^2}{(-4)^2 - 1} = \frac{6 \cdot 16}{16 - 1} = \frac{96}{15} = 6.4 \]

Substitute \( x = -3 \): \[ f(-3) = \frac{6(-3)^2}{(-3)^2 - 1} = \frac{6 \cdot 9}{9 - 1} = \frac{54}{8} = 6.75 \]

Substitute \( x = -\frac{1}{2} \): \[ f\left(-\frac{1}{2}\right) = \frac{6\left(-\frac{1}{2}\right)^2}{\left(-\frac{1}{2}\right)^2 - 1} = \frac{6 \cdot \frac{1}{4}}{\frac{1}{4} - 1} = \frac{\frac{6}{4}}{\frac{1}{4} - 1} = \frac{1.5}{-\frac{3}{4}} = -2 \]

Final Answer