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

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

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

range

f (x) = \frac{x ^{2} + 4}{x}

f (x) \leq - 4 or f (x) \geq 4

Interval Notation

(- \infty, - 4] \cup [4, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x

x^{2} + 4 = y x

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

Discriminant

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

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

Check if the range interval endpoints are included

y = - 4 :

Included

y = 4 :

Included

Therefore the range is

f (x) \leq - 4 or f (x) \geq 4

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

(x - 2)^{2} (x - 1)

f (x) = - 1

f (x) = \frac{5 x + 3}{4 x - 1}

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

What is the range of f(x)=(x^2+4)/x ?
The range of f(x)=(x^2+4)/x is f(x)<=-4\lor f(x)>= 4