What is the range of 1/(x^2-10x+25) ?

The range of 1/(x^2-10x+25) is f(x)>0

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range of 1/(x^2-10x+25)

range

\frac{1}{x ^{2} - 10 x + 25}

f (x) > 0

Interval Notation

(0, \infty)

Solution steps

Rewrite as

\frac{1}{x ^{2} - 10 x + 25} = y

Multiply both sides by

x^{2} - 10 x + 25

1 = y (x^{2} - 10 x + 25)

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

Discriminant

1 = y (x^{2} - 10 x + 25) : 4 y

4 y \geq 0 : y \geq 0

Check if the range interval endpoints are included

y = 0 :

Excluded

Therefore the range is

f (x) > 0

f (x) = x^{2} - 9

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

f (x) = 36 - 9 x^{2}

(- 3, - 6), (1, 4)

- 2 x^{2} + 2 x - 4