What is the range of f(x)=(3x^2)/(x^2+4) ?

The range of f(x)=(3x^2)/(x^2+4) is 0<= f(x)<3

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range of f(x)=(3x^2)/(x^2+4)

range

f (x) = \frac{3 x ^{2}}{x ^{2} + 4}

0 \leq f (x) < 3

Interval Notation

[0, 3)

Solution steps

Rewrite as

\frac{3 x ^{2}}{x ^{2} + 4} = y

Multiply both sides by

x^{2} + 4

3 x^{2} = y (x^{2} + 4)

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

Discriminant

3 x^{2} = y (x^{2} + 4) : - 16 y^{2} + 48 y

- 16 y^{2} + 48 y \geq 0 : 0 \leq y \leq 3

Check if the range interval endpoints are included

y = 0 :

Included

y = 3 :

Excluded

Therefore the range is

0 \leq f (x) < 3

f (x) = (x + 3)

f (x) = 4^{x}

(- 3, 8), (3, - 1)

3 (\frac{2}{3})^{x}

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

What is the range of f(x)=(3x^2)/(x^2+4) ?
The range of f(x)=(3x^2)/(x^2+4) is 0<= f(x)<3