What is the range of (x^2+3x-2)/(x^2+2x-3) ?

The range of (x^2+3x-2)/(x^2+2x-3) is -infinity <f(x)<infinity

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range of (x^2+3x-2)/(x^2+2x-3)

range

\frac{x ^{2} + 3 x - 2}{x ^{2} + 2 x - 3}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{x ^{2} + 3 x - 2}{x ^{2} + 2 x - 3} = y

Multiply both sides by

x^{2} + 2 x - 3

x^{2} + 3 x - 2 = y (x^{2} + 2 x - 3)

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

Discriminant

x^{2} + 3 x - 2 = y (x^{2} + 2 x - 3) : 16 y^{2} - 32 y + 17

16 y^{2} - 32 y + 17 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

x^{3} - 6 x^{2}

f (x) = 4 x^{2} + 7

\frac{3 - x}{x - 2}

y = \frac{1}{2} x + 2

\frac{2 x}{x - 5}

What is the range of (x^2+3x-2)/(x^2+2x-3) ?
The range of (x^2+3x-2)/(x^2+2x-3) is -infinity <f(x)<infinity