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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

range

f (x) = \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

f (x) = x^{2} + 5 x - 14

f (x) = \frac{1}{( x - 3 )}

y + 6 = \frac{1}{3} (x - 4)

f (x) = 3 x^{3}

f (x) = - x^{3} + 3 x^{2} + 9 x + 1

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