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

The range of f(x)=(x-1)/(1+x^2) is -sqrt(1/2)-1/2 <= f(x)<= sqrt(1/2)-1/2

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

range

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

- \frac{1}{2} - \frac{1}{2} \leq f (x) \leq \frac{1}{2} - \frac{1}{2}

Interval Notation

[- \frac{1}{2} - \frac{1}{2}, \frac{1}{2} - \frac{1}{2}]

Solution steps

Rewrite as

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

Multiply both sides by

1 + x^{2}

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

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

Discriminant

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

1 - 4 y^{2} - 4 y \geq 0 : - \frac{1}{2} - \frac{1}{2} \leq y \leq \frac{1}{2} - \frac{1}{2}

Check if the range interval endpoints are included

y = - \frac{1}{2} - \frac{1}{2} :

Included

y = \frac{1}{2} - \frac{1}{2} :

Included

Therefore the range is

- \frac{1}{2} - \frac{1}{2} \leq f (x) \leq \frac{1}{2} - \frac{1}{2}

(- 4, 4), (5, - 1)

x + 3 y = 12

2 x + 3 y = 7

(24, 22), (13, 29)

3^{x}

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