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

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

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

range

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

2 \leq f (x) < 3

Interval Notation

[2, 3)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

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

Inverse of

2 + \frac{x ^{2}}{x ^{2} + 4} : \frac{- 4 x + 8}{x - 3}, - \frac{- 4 x + 8}{x - 3}

\frac{- 4 x + 8}{x - 3}, - \frac{- 4 x + 8}{x - 3}

Find the domain of each inverse function

Domain of

\frac{- 4 x + 8}{x - 3} : 2 \leq x < 3

Domain of

- \frac{- 4 x + 8}{x - 3} : 2 \leq x < 3

Combine the intervals

2 \leq f (x) < 3

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

\frac{- 6}{25 x ^{\frac{8}{5}}}

4 x^{4}

f (x) = - 3

f (x) = - \frac{1}{4} x + \frac{1}{4}

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