What is the range of (5x^2+5)/(x^2+4x+4) ?

The range of (5x^2+5)/(x^2+4x+4) is f(x)>= 1

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range of (5x^2+5)/(x^2+4x+4)

range

\frac{5 x ^{2} + 5}{x ^{2} + 4 x + 4}

f (x) \geq 1

Solution steps

Rewrite as

\frac{5 x ^{2} + 5}{x ^{2} + 4 x + 4} = y

Multiply both sides by

x^{2} + 4 x + 4

5 x^{2} + 5 = y (x^{2} + 4 x + 4)

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

Discriminant

5 x^{2} + 5 = y (x^{2} + 4 x + 4) : 100 y - 100

100 y - 100 \geq 0 : y \geq 1

Check if the range interval endpoints are included

y = 1 :

Included

Therefore the range is

f (x) \geq 1

x - 3 + 2

(- 5, 3), (0, - 1)

f (x) = \frac{1}{3} x + 5

f (x) = x^{3} - 3 x^{2} + 1

y = \frac{1}{- 4 - 3}

What is the range of (5x^2+5)/(x^2+4x+4) ?
The range of (5x^2+5)/(x^2+4x+4) is f(x)>= 1