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

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

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

range

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

Multiply both sides by

x^{2} - 4

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

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

Discriminant

x^{2} + 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

y = 6 x - 7

\frac{4 x}{x - 1}

f (x) = 4 x - 2

- x^{2} + 5 x - 7

Y (x) = \frac{2}{3} x + 3

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