What is the domain and range of $\frac{x + 5}{x^{2} + 36}$ $(x+5)/(x^2+36)$ ?

Question

2245 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-15T13:13:18

The domain is $x in RR$ .
The range is $y in [-0.04,0.18]$

The denominator is $>0$

$AA x in RR$ , $x^2+36>0$

Therefore,

The domain is $x in RR$

Let,

$y=(x+5)/(x^2+36)$

Simplifying and rearranging

$y(x^2+36)=x+5$

$yx^2-x+36y-5=0$

This is a quadratic equation in $x^2$

In order for this equation to have solutions, the discriminant $Delta>=0$

So,

$Delta=b^2-4ac=(-1)^2-4(y)(36y-5)>=0$

$1-144y^2+20y>=0$

$144y^2-20y-1<=0$

$y=(20+-sqrt(400+4*144))/(288)$

$y_1=(20+31.24)/188=0.18$

$y_2=(20-31.24)/288=-0.04$

Therefore,

The range is $y in [-0.04,0.18]$

graph{(x+5)/(x^2+36) [-8.89, 8.884, -4.44, 4.44]}

What is the domain and range of (x+5)/(x^2+36)x+5x2+36(x+5)/(x^2+36)?