What is the range of f(x)=(3x^2+2x-1)/(6x^2-7x-3) ?

The range of f(x)=(3x^2+2x-1)/(6x^2-7x-3) is -infinity <f(x)<infinity

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range of f(x)=(3x^2+2x-1)/(6x^2-7x-3)

range

f (x) = \frac{3 x ^{2} + 2 x - 1}{6 x ^{2} - 7 x - 3}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{3 x ^{2} + 2 x - 1}{6 x ^{2} - 7 x - 3} = y

Multiply both sides by

6 x^{2} - 7 x - 3

3 x^{2} + 2 x - 1 = y (6 x^{2} - 7 x - 3)

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

Discriminant

3 x^{2} + 2 x - 1 = y (6 x^{2} - 7 x - 3) : 121 y^{2} - 32 y + 16

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

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

f (x) = 2 x^{3} - 113

y = 2 x^{2} - 12 x - 2

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

f (x) = 7 x^{3} + 3

\frac{x ^{4}}{x ^{2} + x - 6}

What is the range of f(x)=(3x^2+2x-1)/(6x^2-7x-3) ?
The range of f(x)=(3x^2+2x-1)/(6x^2-7x-3) is -infinity <f(x)<infinity