What is the range of (x-3)/(x^2-16) ?

The range of (x-3)/(x^2-16) is -infinity <f(x)<infinity

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range of (x-3)/(x^2-16)

range

\frac{x - 3}{x ^{2} - 16}

- \infty < f (x) < \infty

Solution steps

Rewrite as

\frac{x - 3}{x ^{2} - 16} = y

Multiply both sides by

x^{2} - 16

x - 3 = y (x^{2} - 16)

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

Discriminant

x - 3 = y (x^{2} - 16) : 1 + 64 y^{2} - 12 y

1 + 64 y^{2} - 12 y \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

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

f (x) = 32 - 4 x

(- 2, 4), (2.5, 3.5)

f (x) = - 3 sin (\frac{1}{2} x + \frac{π}{5})

3 x + 1

What is the range of (x-3)/(x^2-16) ?
The range of (x-3)/(x^2-16) is -infinity <f(x)<infinity