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

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Answer 1 · 2018-06-26T06:31:27

The domain is $=RR$ .
The range is $y in [-0.4,1.24 ]$

The function is

$f(x)=(x+6)/(x^2+5)$

$AA x in RR$ , the denominator is $x^2+5>0$

The domain is $=RR$

To calculate the range, let

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

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

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

This is a quadratic equation in $x^2$ and in order to have solutions, the discriminant must be $>=0$

Therefore,

$Delta=(-1)^2-4(y)(5y-6)>=0$

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

$=>$ , $20y^2-24y-1<=0$

The solutions to this inequality is

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

$y in [-0.4,1.24 ]$

The range is $y in [-0.4,1.24 ]$

graph{(x+6)/(x^2+5) [-13.7, 14.78, -7.27, 6.97]}

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