To express [tex]\( f(x) = |x - 6| \)[/tex] as a piecewise function, observe how the absolute value function behaves differently based on the input values for [tex]\( x \)[/tex] relative to 6. Specifically, within the function [tex]\( |x - 6| \)[/tex]:
1. When [tex]\( x \geq 6 \)[/tex], the expression [tex]\( x - 6 \)[/tex] is non-negative, so the absolute value function behaves as [tex]\( x - 6 \)[/tex].
2. When [tex]\( x < 6 \)[/tex], the expression [tex]\( x - 6 \)[/tex] is negative, so the absolute value function behaves as [tex]\(-(x - 6)\)[/tex] which simplifies to [tex]\( -x + 6 \)[/tex].
So we deal with two cases:
- For [tex]\( x \geq 6 \)[/tex], [tex]\( f(x) = x - 6 \)[/tex]
- For [tex]\( x < 6 \)[/tex], [tex]\( f(x) = -x + 6 \)[/tex]
Thus, the piecewise function for [tex]\( f(x) \)[/tex] looks like this:
[tex]\[ f(x) = \begin{cases}
x - 6 & \text{if } x \geq 6 \\
-x + 6 & \text{if } x < 6
\end{cases}
\][/tex]
Given the options provided:
1. [tex]\( f(x)=\left\{\begin{array}{c}x-6, x \geq 0 \\ -x-6, x<0\end{array}\right. \)[/tex]
2. [tex]\( f(x)=\left\{\begin{array}{c}x-6, x \geq 6 \\ -x+6, x<6\end{array}\right. \)[/tex]
3. [tex]\( f(x)=\left\{\begin{array}{l}x-6, x \geq 0 \\ -x+6, x<0\end{array}\right. \)[/tex]
4. [tex]\( f(x)=\left\{\begin{array}{c}x-6, x \geq 6 \\ -x-6, x<6\end{array}\right. \)[/tex]
The correct piecewise representation is:
[tex]\[ f(x)=\left\{\begin{array}{c}x-6, x \geq 6 \\ -x+6, x<6\end{array}\right. \][/tex]
So, the correct answer is the second option.
