What is the range of f(x)=(2+x^2)/(x^2-4) ?

The range of f(x)=(2+x^2)/(x^2-4) is f(x) 1

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range of f(x)=(2+x^2)/(x^2-4)

range

f (x) = \frac{2 + x ^{2}}{x ^{2} - 4}

f (x) \leq - \frac{1}{2} or f (x) > 1

Interval Notation

(- \infty, - \frac{1}{2}] \cup (1, \infty)

Solution steps

Rewrite as

\frac{2 + x ^{2}}{x ^{2} - 4} = y

Multiply both sides by

x^{2} - 4

2 + x^{2} = y (x^{2} - 4)

The range is the set of y for which the discriminant is greater or equal to zero

Discriminant

2 + x^{2} = y (x^{2} - 4) : 16 y^{2} - 8 y - 8

16 y^{2} - 8 y - 8 \geq 0 : y \leq - \frac{1}{2} or y \geq 1

Check if the range interval endpoints are included

y = - \frac{1}{2} :

Included

y = 1 :

Excluded

Therefore the range is

f (x) \leq - \frac{1}{2} or f (x) > 1

\frac{2}{3 ( x + 5 ) ^{\frac{1}{3}}}

- 2 sin (2 x + \frac{2 π}{5}) + 3

f (x) = \frac{5 x + 1}{2 x ^{2} - 2 x - 12}

y = 0.8 x + 8.7

g (x) = - \frac{1}{2 6 - x}

What is the range of f(x)=(2+x^2)/(x^2-4) ?
The range of f(x)=(2+x^2)/(x^2-4) is f(x)<=-1/2 \lor f(x)>1