Question

Question #99878

Answer 1

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)`

Explanation:

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]}