What is the range of f(x)=(5x^2+20x+23)/(x^2+4x+5) ?

The range of f(x)=(5x^2+20x+23)/(x^2+4x+5) is 3<= f(x)<5

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range of f(x)=(5x^2+20x+23)/(x^2+4x+5)

range

f (x) = \frac{5 x ^{2} + 20 x + 23}{x ^{2} + 4 x + 5}

3 \leq f (x) < 5

Interval Notation

Solution steps

Rewrite as

\frac{5 x ^{2} + 20 x + 23}{x ^{2} + 4 x + 5} = y

Multiply both sides by

x^{2} + 4 x + 5

5 x^{2} + 20 x + 23 = y (x^{2} + 4 x + 5)

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

Discriminant

5 x^{2} + 20 x + 23 = y (x^{2} + 4 x + 5) : - 4 y^{2} + 32 y - 60

- 4 y^{2} + 32 y - 60 \geq 0 : 3 \leq y \leq 5

Check if the range interval endpoints are included

y = 3 :

Included

y = 5 :

Excluded

Therefore the range is

3 \leq f (x) < 5

\frac{3 - x ^{2}}{x ^{2} - 4}

y = (x + 1)^{2} - 9

f (x) = 3

3 - e^{x}

lo g_{8} (x)

What is the range of f(x)=(5x^2+20x+23)/(x^2+4x+5) ?
The range of f(x)=(5x^2+20x+23)/(x^2+4x+5) is 3<= f(x)<5