What is the range of f(x)= 1/(x^2-6x+11) ?

The range of f(x)= 1/(x^2-6x+11) is 0<f(x)<= 1/2

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range of f(x)= 1/(x^2-6x+11)

range

f (x) = \frac{1}{x ^{2} - 6 x + 11}

0 < f (x) \leq \frac{1}{2}

Interval Notation

(0, \frac{1}{2}]

Solution steps

Rewrite as

\frac{1}{x ^{2} - 6 x + 11} = y

Multiply both sides by

x^{2} - 6 x + 11

1 = y (x^{2} - 6 x + 11)

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

Discriminant

1 = y (x^{2} - 6 x + 11) : - 8 y^{2} + 4 y

- 8 y^{2} + 4 y \geq 0 : 0 \leq y \leq \frac{1}{2}

Check if the range interval endpoints are included

y = 0 :

Excluded

y = \frac{1}{2} :

Included

Therefore the range is

0 < f (x) \leq \frac{1}{2}

What is the range of f(x)= 1/(x^2-6x+11) ?
The range of f(x)= 1/(x^2-6x+11) is 0<f(x)<= 1/2