# Question #99878

Oct 10, 2017

The domain of $f \left(x\right)$ is $x \in \mathbb{R}$ and the range is $f \left(x\right) \in \left(- \infty , 12.5\right]$
The domain of $f \left(x\right)$ is $x \in \mathbb{R}$ and the range is $g \left(x\right) \in \left[0 , + \infty\right)$

#### Explanation:

First part $f \left(x\right)$

$f \left(x\right) = 12 - 2 x - 2 {x}^{2}$

This is a polynomial function and it is defined over $\mathbb{R}$

Therefore,

The domain of $f \left(x\right)$ is $x \in \mathbb{R}$

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

Rewriting the equation

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

For this quadratic equation to have solutions, the discriminant $\Delta \ge 0$

$\Delta = {b}^{2} - 4 a c = {\left(2\right)}^{2} - 4 \cdot \left(2\right) \cdot \left(y - 12\right)$

$4 - 8 y + 96 \ge 0$

$8 y \le 100$

$y \le \frac{100}{8}$

The range is $y \in \left(- \infty , 12.5\right]$

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

Second part *$g \left(x\right)$*

$g \left(x\right) = | f \left(x\right) |$

The domain of $g \left(x\right)$ remains the same, $x \in \mathbb{R}$

The range changes as all the negative values become positive

So,

the range is $g \left(x\right) \in \left[0 , + \infty\right)$

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