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

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

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

range

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

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{x - 3}{x ^{2} - 5 x} = y

Multiply both sides by

x^{2} - 5 x

x - 3 = y (x^{2} - 5 x)

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

Discriminant

x - 3 = y (x^{2} - 5 x) : 25 y^{2} - 2 y + 1

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

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

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

(9 - x^{\frac{2}{3}})^{\frac{3}{2}}

f (x) = 2 x^{2} - 10 x + 12

f (x) = \frac{4 x}{x - 2}

3 \cdot 1.3 5^{x}

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