Final answer:
The range of the given expression, 2(x² - 9)/(x² - 4), is (-∞, 0) ∪ (0, +∞).
Step-by-step explanation:
The range of the given expression, 2(x² - 9)/(x² - 4), can be found by plotting points on a graph and observing the behavior of the function. To do this, we can find the vertical asymptotes and the behavior of the function as x approaches the asymptotes. The vertical asymptotes occur when the denominator of the function is equal to zero, so we solve x² - 4 = 0 to find the values at which the function is undefined. The solutions to this equation are x = 2 and x = -2. These values are the vertical asymptotes of the function.
Now let's observe the behavior of the function as x approaches these asymptotes. We can do this by plugging in values that are very close to the asymptotes, both greater and smaller. For x values greater than 2, the expression becomes positive infinity. For x values between -2 and 2, the expression takes negative values but approaches negative infinity as x approaches -2. For x values smaller than -2, the expression becomes positive infinity again. This means that the range of the function is (-∞, 0) ∪ (0, +∞).