What is the range of f(x)=(x^3-3x^2-4x)/(x-4) ?

The range of f(x)=(x^3-3x^2-4x)/(x-4) is f(x)>=-1/4

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range of f(x)=(x^3-3x^2-4x)/(x-4)

range

f (x) = \frac{x ^{3} - 3 x ^{2} - 4 x}{x - 4}

f (x) \geq - \frac{1}{4}

Interval Notation

[- \frac{1}{4}, \infty)

Solution steps

Find the minimum and maximum value in each defined interval and unite the results

Domain of

\frac{x ^{3} - 3 x ^{2} - 4 x}{x - 4} : x < 4 or x > 4

Simplify

\frac{x ^{3} - 3 x ^{2} - 4 x}{x - 4} : x (x + 1)

Extreme Points of

x (x + 1) :

Minimum

(- \frac{1}{2}, - \frac{1}{4})

Find the range for the interval

- \infty < x < 4 : - \frac{1}{4} \leq f (x) < \infty

Find the range for the interval

4 < x < \infty : 20 < f (x) < \infty

Combine the ranges of all domain intervals to obtain the function range

f (x) \geq - \frac{1}{4} or f (x) > 20

f (x) \geq - \frac{1}{4}

What is the range of f(x)=(x^3-3x^2-4x)/(x-4) ?
The range of f(x)=(x^3-3x^2-4x)/(x-4) is f(x)>=-1/4