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

The range of (x^2)/(x^2+4) is 0<= f(x)<1

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range of (x^2)/(x^2+4)

range

\frac{x ^{2}}{x ^{2} + 4}

0 \leq f (x) < 1

Interval Notation

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + 4

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

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

Discriminant

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

- 16 y^{2} + 16 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) = \frac{x + 17}{x - 16}

h (x) = (x + 1)^{3} + 3

f (x) = 2 ∣ x ∣ - 3

\frac{7}{x + 2}

f (x) = 5 x