What is the range of f(x)=(x^2+6x+11)/(2x^2+12x+18) ?

The range of f(x)=(x^2+6x+11)/(2x^2+12x+18) is f(x)> 1/2

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range of f(x)=(x^2+6x+11)/(2x^2+12x+18)

range

f (x) = \frac{x ^{2} + 6 x + 11}{2 x ^{2} + 12 x + 18}

f (x) > \frac{1}{2}

Interval Notation

(\frac{1}{2}, \infty)

Solution steps

Rewrite as

\frac{x ^{2} + 6 x + 11}{2 x ^{2} + 12 x + 18} = y

Multiply both sides by

2 x^{2} + 12 x + 18

x^{2} + 6 x + 11 = y (2 x^{2} + 12 x + 18)

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

Discriminant

x^{2} + 6 x + 11 = y (2 x^{2} + 12 x + 18) : 16 y - 8

16 y - 8 \geq 0 : y \geq \frac{1}{2}

Check if the range interval endpoints are included

y = \frac{1}{2} :

Excluded

Therefore the range is

f (x) > \frac{1}{2}

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

f (x) = 7 tan (0.4 x)

f (x) = \frac{2 x}{x - 1}

f (x) = \frac{- x + 6}{x ^{2} - 49}

2 y - 4 x = - 18

What is the range of f(x)=(x^2+6x+11)/(2x^2+12x+18) ?
The range of f(x)=(x^2+6x+11)/(2x^2+12x+18) is f(x)> 1/2