What is the f(x)=(3x^2+27x+54)/(x^2-10x) ?

The f(x)=(3x^2+27x+54)/(x^2-10x) is

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f(x)=(3x^2+27x+54)/(x^2-10x)

f (x) = \frac{3 x ^{2} + 27 x + 54}{x ^{2} - 10 x}

Domain : x < 0 or 0 < x < 10 or x > 10

Range : f (x) \leq - \frac{18 26}{25} - \frac{189}{50} or f (x) \geq \frac{18 26}{25} - \frac{189}{50}

X Intercepts : (- 3, 0), (- 6, 0)

Asymptotes : Vertical x = 0, x = 10, Horizontal y = 3

Extreme Points : Minimum (- \frac{6 ( 3 + 2 26 )}{19}, - \frac{9 ( 21 - 4 26 )}{50}), Maximum (\frac{6 ( 2 26 - 3 )}{19}, - \frac{9 ( 21 + 4 26 )}{50})

Interval Notation

Domain : (- \infty, 0) \cup (0, 10) \cup (10, \infty)

Range : (- \infty, - \frac{18 26}{25} - \frac{189}{50}] \cup [\frac{18 26}{25} - \frac{189}{50}, \infty)

Solution steps

Domain of

\frac{3 x ^{2} + 27 x + 54}{x ^{2} - 10 x} : x < 0 or 0 < x < 10 or x > 10

Range of

\frac{3 x ^{2} + 27 x + 54}{x ^{2} - 10 x} : f (x) \leq - \frac{18 26}{25} - \frac{189}{50} or f (x) \geq \frac{18 26}{25} - \frac{189}{50}

Axis interception points of

\frac{3 x ^{2} + 27 x + 54}{x ^{2} - 10 x} :

X Intercepts

: (- 3, 0), (- 6, 0)

Asymptotes of

\frac{3 x ^{2} + 27 x + 54}{x ^{2} - 10 x} :

Vertical

: x = 0, x = 10,

Horizontal

: y = 3

Extreme Points of

\frac{3 x ^{2} + 27 x + 54}{x ^{2} - 10 x} :

Minimum

(- \frac{6 ( 3 + 2 26 )}{19}, - \frac{9 ( 21 - 4 26 )}{50}),

Maximum

(\frac{6 ( 2 26 - 3 )}{19}, - \frac{9 ( 21 + 4 26 )}{50})

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