How do you find domain and range for $f(x)=(x+6)/(x^2+5)$ ?

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Answer 1 · 2018-05-13T13:34:24

The domain is $x in RR$ . The range is $y in[-0.04, 1.24]$

The denominator is always $>0$ whatever $x in RR$

The domain is $x in RR$

To find the domain, proceed as follows

Let $y=(x+6)/(x^2+5)$

Rearranging,

$y(x^2+5)=x+6$

$yx^2-x+5y-6=0$

This is a quadratic equation in $x$ and in order for this equation to have solutions, the discriminant $>=0$

$Delta=(-1)^2-4*y(5y-6)=1-20y^2+24y$

$-20y^2+24y+1>=0$

The solutions to this inequality is

$y in [(-24+sqrt(24^2+4*20))/(-40),(-24-sqrt(24^2+4*20))/(-40)]$

$y in[-0.04, 1.24]$

The range is $y in[-0.04, 1.24]$

graph{(x+6)/(x^2+5) [-10.47, 3.574, -3.903, 3.117]}

How do you find domain and range for f(x)=(x+6)/(x^2+5) f(x)=(x+6)/(x^2+5) ?