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

The range of (3x)/((x+2)^2) is f(x)<= 3/8

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range of (3x)/((x+2)^2)

range

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

f (x) \leq \frac{3}{8}

Interval Notation

(- \infty, \frac{3}{8}]

Solution steps

Rewrite as

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

Multiply both sides by

(x + 2)^{2}

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

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

Discriminant

3 x = y (x + 2)^{2} : - 24 y + 9

- 24 y + 9 \geq 0 : y \leq \frac{3}{8}

Check if the range interval endpoints are included

y = \frac{3}{8} :

Included

Therefore the range is

f (x) \leq \frac{3}{8}

f (x) = x + 8 - x

f (x) = x - 7 - 7

f (x) = 3.1 + 6.6 x - 2.3 x^{2}

\frac{4 x + 7}{3 x - 4}

f (x) = 3 x^{2} + 2 x - 6