What is the range of f(x)=(2x-5)/(x(x-3)) ?

The range of f(x)=(2x-5)/(x(x-3)) is -infinity <f(x)<infinity

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range of f(x)=(2x-5)/(x(x-3))

range

f (x) = \frac{2 x - 5}{x ( x - 3 )}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{2 x - 5}{x ( x - 3 )} = y

Multiply both sides by

x (x - 3)

2 x - 5 = y x (x - 3)

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

Discriminant

2 x - 5 = y x (x - 3) : 9 y^{2} - 8 y + 4

9 y^{2} - 8 y + 4 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

y - 2 = 3 (x - 1)

f (x) = \frac{3 x + 7}{2 x + 1}

6 x^{2} + 12 x

f (x) = \frac{( x - 2 )}{x - 3}

y = 5 csc (8 x^{4} - 2 x + 1)

What is the range of f(x)=(2x-5)/(x(x-3)) ?
The range of f(x)=(2x-5)/(x(x-3)) is -infinity <f(x)<infinity