How to find the range of this function?

Question

f(x) = x^2-1/x^2-4 (it is a rational function) however, unlike some rational functions, the range is just the value of horizontal asymptote. In this case, the horizontal asymptote is y=1, but that does not satisfy the range. How do I find the range?

and does the same method apply if the denominator of the rational function equals all real numbers?

Answer 1

Hi Ariel G.

(x²-1)/(x²-4)

Can be factored

(x + 1)(x-1)/(x +2)(x-2)

x ≠ +/-2 this would result in division by zero in the denominator so those would be barriers.

You would have vertical asymptotes at 2 and -2. .

I hope this helps you with the range

You can graph the function at Desmos.com to see your y values. .It was difficult to get the proper enclosures in at this website I hope they show up clearly on your end.

(-∞, 1/4⌉∪(1,∞),{y¦y<=1/4, y>1}

Answer 2

Range of this function: - ∞ < y < ∞, because lim f(x) x→0 = - ∞ and lim f(x) x→±∞ = ∞ and no horizontal

asymptote.