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

The range of f(x)=(2x^2+2x)/(x^2-4x+4) is f(x)>=-1/12

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

range

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

f (x) \geq - \frac{1}{12}

Interval Notation

[- \frac{1}{12}, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} - 4 x + 4

2 x^{2} + 2 x = y (x^{2} - 4 x + 4)

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

Discriminant

2 x^{2} + 2 x = y (x^{2} - 4 x + 4) : 48 y + 4

48 y + 4 \geq 0 : y \geq - \frac{1}{12}

Check if the range interval endpoints are included

y = - \frac{1}{12} :

Included

Therefore the range is

f (x) \geq - \frac{1}{12}

x x - 4

x^{3}

y = \frac{4 x + 2}{3}

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

f (x) = x^{2} + 6

What is the range of f(x)=(2x^2+2x)/(x^2-4x+4) ?
The range of f(x)=(2x^2+2x)/(x^2-4x+4) is f(x)>=-1/12