What is the range of 5/(x^2+1) ?

The range of 5/(x^2+1) is 0<f(x)<= 5

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range of 5/(x^2+1)

range

\frac{5}{x ^{2} + 1}

0 < f (x) \leq 5

Interval Notation

(0, 5]

Solution steps

Rewrite as

\frac{5}{x ^{2} + 1} = y

Multiply both sides by

x^{2} + 1

5 = y (x^{2} + 1)

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

Discriminant

5 = y (x^{2} + 1) : - 4 y^{2} + 20 y

- 4 y^{2} + 20 y \geq 0 : 0 \leq y \leq 5

Check if the range interval endpoints are included

y = 0 :

Excluded

y = 5 :

Included

Therefore the range is

0 < f (x) \leq 5

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

f (x) = 6 - x

f (x) = 3 x^{2} + 6

Y = 2 X^{2} - 3

(2 x - 3)^{2}