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

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

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

range

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

f (x) \leq \frac{2}{9} or f (x) \geq 2

Interval Notation

(- \infty, \frac{2}{9}] \cup [2, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + x - 2

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

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

Discriminant

2 x - 4 = y (x^{2} + x - 2) : 9 y^{2} - 20 y + 4

9 y^{2} - 20 y + 4 \geq 0 : y \leq \frac{2}{9} or y \geq 2

Check if the range interval endpoints are included

y = \frac{2}{9} :

Included

y = 2 :

Included

Therefore the range is

f (x) \leq \frac{2}{9} or f (x) \geq 2

f (x) = 2 x^{3} + 3 x^{2} - 12 x + 5

f (x) = 5 x^{3} - 9

y = \frac{1}{5} x + \frac{4}{5}, (1, 1)

x^{2} + 2

f (x) = x^{4} - 7 x^{3} + 21 x^{2} - 23 x - 52

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