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

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

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range of x/(x^2+4)

range

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

- \frac{1}{4} \leq f (x) \leq \frac{1}{4}

Interval Notation

[- \frac{1}{4}, \frac{1}{4}]

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + 4

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

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

Discriminant

x = y (x^{2} + 4) : 1 - 16 y^{2}

1 - 16 y^{2} \geq 0 : - \frac{1}{4} \leq y \leq \frac{1}{4}

Check if the range interval endpoints are included

y = - \frac{1}{4} :

Included

y = \frac{1}{4} :

Included

Therefore the range is

- \frac{1}{4} \leq f (x) \leq \frac{1}{4}

4 tan (2 x + π) + 1

f (x) = 3 x + x - 2

f (x) = - \frac{1}{81} x^{2} + 100

4 x - 2 y = 8

f (x) = (x - 3)^{2} + 5