Question

What is the range of f of x equals StartFraction 2 x squared minus 2 x minus 4 over x squared minus 4 EndFraction?

Answer options with 5 options
A.
All real numbers except negative 2.

B.
All real numbers except 2.

C.
All real numbers except three-halves and 2.

D.
All real numbers except three-halves​ and negative 2.

E.
All real numbers except 2 and negative 2.

Answer 1

Factor numerator: 2x^2 - 2x - 4 = 2(x-2)(x+1). So for x ≠ ±2,
f(x) = 2(x+1)/(x+2).

Rewrite f(x) = 2 - 2/(x+2). As x varies over R\{−2} this takes every real value except 2 (since 2/(x+2) never equals 0). The value 3/2 would occur only when 2(x+1)/(x+2)=3/2, which gives x=2, but x=2 is not in the domain (hole). So 3/2 is also not attained.

Therefore the range is all real numbers except 3/2 and 2. Answer: C.