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

The range of (x^2)/(x^2+1) is 0<= f(x)<1

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

range

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

0 \leq f (x) < 1

Interval Notation

[0, 1)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + 1

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

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

Discriminant

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

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

Check if the range interval endpoints are included

y = 0 :

Included

y = 1 :

Excluded

Therefore the range is

0 \leq f (x) < 1

2 x + 4

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

4 + 7 25 - x^{2}

m = 2

f (x) = 4 x - 1