Question #99878

Question

1250 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-10-10T13:09:58

The domain of $f(x)$ is $x in RR$ and the range is $f(x) in (-oo,12.5]$
The domain of $f(x)$ is $x in RR$ and the range is $g(x) in [0,+oo)$

First part $f(x)$

$f(x)=12-2x-2x^2$

This is a polynomial function and it is defined over $RR$

Therefore,

The domain of $f(x)$ is $x in RR$

Let $y=12-2x-2x^2$

Rewriting the equation

$2x^2+2x-12+y=0$

For this quadratic equation to have solutions, the discriminant $Delta>=0$

$Delta=b^2-4ac=(2)^2-4*(2)*(y-12)$

$4-8y+96>=0$

$8y<=100$

$y<=100/8$

The range is $y in (-oo,12.5]$

graph{12-2x-2x^2 [-26.34, 31.37, -7.27, 21.6]}

Second part * $g(x)$ *

$g(x)=|f(x)|$

The domain of $g(x)$ remains the same, $x in RR$

The range changes as all the negative values become positive

So,

the range is $g(x) in [0,+oo)$

graph{|12-2x-2x^2| [-29.1, 28.6, -1.96, 26.9]}