Question

Convert the following absolute value function into piece wise function.
$g\left(x\right)=|x^2-4|$

• A. $\{\begin{array}{cc}{x}^2+4& ifx\le -2orx\ge 2\\ 4-x^2& if-2<x<2\end{array}$
• B. $\{\begin{array}{cc}{x}^2-4& ifx\le -2orx\ge 2\\ 4-x^2& if-2<x<2\end{array}$
• C. $\{\begin{array}{cc}{x}^2-4& ifx\le -2orx\ge 2\\ 4+x^2& if-2<x<2\end{array}$
• D. None of these

Answer 1

The correct option is B $\{\begin{array}{cc}{x}^2-4& ifx\le -2orx\ge 2\\ 4-x^2& if-2<x<2\end{array}$

$|x^2-4|=x^2-4$ when $x^2\ge 4$

$=4-x^2$ when $x^2<4$

The above 2 intervals can be reduced to

$|x|\ge 2\Rightarrow x\le -2$ or $x\ge 2$

$|x|<2\Rightarrow -2<x<2$