Find the domain and vertical asymptotes

Answer 1

Hello.

The vertical asymptotes are present anytime a value of x causes the function to approach positive or negative ∞, which happens when the function in the denominator equals 0

x² -4 = 0

(x +2) (x -2) = 0

x = 2, x = -2

There are two vertical asymptotes: one at x = -2 and one at x = 2

As for the domain, this is simply the range of x values which the function covers. In this case. there are no other values of x which are impossible. Thus, the domain is (-∞, -2) and (-2,2) and (2,∞). In other words, all real numbers except -2 and +2.

Answer 2

Luckily for us, we only have to look at one thing to find both. The domain is our input. What x-values are we allowed to use? Well, we can't have the denominator equal 0.

x² - 4 = 0

That's the difference of two squares, x² and 4. Since the square root of x² is x and the square root of 4 is 2, we have...

(x + 2)(x - 2) = 0

x + 2 = 0

x = -2

x - 2 = 0

x = 2

Our domain is x≠ -2, 2 and our vertical asymptotes are x = -2 and x = 2.

Hope this helps!