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

The range of 4/(x^2+x-2) is f(x) 0

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

range

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

f (x) \leq - \frac{16}{9} or f (x) > 0

Interval Notation

(- \infty, - \frac{16}{9}] \cup (0, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + x - 2

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

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

Discriminant

4 = y (x^{2} + x - 2) : 9 y^{2} + 16 y

9 y^{2} + 16 y \geq 0 : y \leq - \frac{16}{9} or y \geq 0

Check if the range interval endpoints are included

y = - \frac{16}{9} :

Included

y = 0 :

Excluded

Therefore the range is

f (x) \leq - \frac{16}{9} or f (x) > 0

f (x) = \frac{1 + x ^{2}}{4}

\frac{x}{x + 1}

(6, 3), (10, - 7)

f (x) = \frac{3 x + 5}{7 x + 4}

f (x) = x + 4 - x

What is the range of 4/(x^2+x-2) ?
The range of 4/(x^2+x-2) is f(x)<=-16/9 \lor f(x)>0