What is the range of (2x-1)/(x^2-1) ?

The range of (2x-1)/(x^2-1) is -infinity <f(x)<infinity

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range of (2x-1)/(x^2-1)

range

\frac{2 x - 1}{x ^{2} - 1}

- \infty < f (x) < \infty

Solution steps

Rewrite as

\frac{2 x - 1}{x ^{2} - 1} = y

Multiply both sides by

x^{2} - 1

2 x - 1 = y (x^{2} - 1)

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

Discriminant

2 x - 1 = y (x^{2} - 1) : 4 + 4 y^{2} - 4 y

4 + 4 y^{2} - 4 y \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

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

f (x) = (4 x + 1)^{3} (4 x^{2} - 4 x + 1)

- x + 3

y = x^{3} + y^{3} - x - y - 2 = 0

What is the range of (2x-1)/(x^2-1) ?
The range of (2x-1)/(x^2-1) is -infinity <f(x)<infinity