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

The range of (x-3)/((x+4)^2) is f(x)<= 1/28

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range of (x-3)/((x+4)^2)

range

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

f (x) \leq \frac{1}{28}

Interval Notation

(- \infty, \frac{1}{28}]

Solution steps

Rewrite as

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

Multiply both sides by

(x + 4)^{2}

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

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

Discriminant

x - 3 = y (x + 4)^{2} : - 28 y + 1

- 28 y + 1 \geq 0 : y \leq \frac{1}{28}

Check if the range interval endpoints are included

y = \frac{1}{28} :

Included

Therefore the range is

f (x) \leq \frac{1}{28}

f (x) = \frac{x + 4}{x - 6}

- 3 x^{2} + 12 x - 4

\frac{1}{x ^{3}}

f (x) = ln (x - 2) + 4

What is the range of (x-3)/((x+4)^2) ?
The range of (x-3)/((x+4)^2) is f(x)<= 1/28