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Consider this absolute value function:
[tex]f(x)=|x-2|[/tex]

If function [tex]f[/tex] is written as a piecewise function, which piece will it include?

A. [tex]-x+2, \quad x\ \textless \ 2[/tex]
B. [tex]-x-2, \quad x \geq 2[/tex]
C. [tex]x+2, \quad x\ \textgreater \ 2[/tex]
D. [tex]x-2, \quad x \leq 2[/tex]



Answer :

GinnyAnswer
To express the absolute value function [tex]\( f(x) = |x - 2| \)[/tex] as a piecewise function, we need to consider the definition of the absolute value function.

The absolute value function [tex]\( |a| \)[/tex] is defined as:
[tex]\[ |a| = \begin{cases} a, & \text{if } a \geq 0 \\ -a, & \text{if } a < 0 \end{cases} \][/tex]

Here, [tex]\( a = x - 2 \)[/tex]. So we need to determine in which cases [tex]\( x - 2 \)[/tex] is non-negative (greater than or equal to zero) and in which cases it is negative (less than zero).

1. When [tex]\( x - 2 \geq 0 \)[/tex]:
[tex]\[ x \geq 2 \][/tex]
In this case, [tex]\( |x - 2| = x - 2 \)[/tex].

2. When [tex]\( x - 2 < 0 \)[/tex]:
[tex]\[ x < 2 \][/tex]
In this case, [tex]\( |x - 2| = -(x - 2) = -x + 2 \)[/tex].

Putting this information together, the piecewise function for [tex]\( f(x) = |x - 2| \)[/tex] is:
[tex]\[ f(x) = \begin{cases} -x + 2, & \text{if } x < 2 \\ x - 2, & \text{if } x \geq 2 \end{cases} \][/tex]

Therefore, the correct pieces that the function will include are:
[tex]\[ -x + 2 \text{ for } x < 2 \][/tex]
and
[tex]\[ x - 2 \text{ for } x \geq 2 \][/tex]

So, the correct answer is:
[tex]\[ -x + 2, \; x < 2 \][/tex]
and
[tex]\[ x - 2, \; x \geq 2 \][/tex]

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