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Consider this absolute value function:
[tex]\[ f(x) = |x + 3| \][/tex]

How can function [tex]\( f \)[/tex] be rewritten as a piecewise function?
[tex]\[
\begin{array}{c}
x \ \textless \ -3 \quad -x + 3 \quad x - 3 \quad -x - 3 \quad x \ \textless \ 3 \quad x \geq -3 \quad x \geq 3 \\
f(x) = \left\{ \begin{array}{ll}
-(x + 3) & \text{if } x \ \textless \ -3 \\
x + 3 & \text{if } x \geq -3
\end{array} \right.
\end{array}
\][/tex]

Sagot :

GinnyAnswer
To rewrite the absolute value function [tex]\( f(x) = |x+3| \)[/tex] as a piecewise function, we need to consider the definition of the absolute value and how it affects the expression based on the input value of [tex]\(x\)[/tex].

The absolute value function [tex]\( |x+3| \)[/tex] has different expressions based on whether [tex]\( x+3 \)[/tex] is non-negative or negative:

1. When [tex]\( x + 3 \)[/tex] is non-negative (i.e., [tex]\( x + 3 \geq 0 \)[/tex]), the absolute value function [tex]\( |x+3| \)[/tex] is simply [tex]\( x + 3 \)[/tex].
2. When [tex]\( x + 3 \)[/tex] is negative (i.e., [tex]\( x + 3 < 0 \)[/tex]), the absolute value function [tex]\( |x+3| \)[/tex] is [tex]\( -(x + 3) \)[/tex], which simplifies to [tex]\( -x - 3 \)[/tex].

To determine the conditions under which each expression applies:
- [tex]\( x + 3 \geq 0 \)[/tex] simplifies to [tex]\( x \geq -3 \)[/tex]
- [tex]\( x + 3 < 0 \)[/tex] simplifies to [tex]\( x < -3 \)[/tex]

Given these conditions, we can write the piecewise function as follows:

[tex]\[ f(x) = \left\{\begin{array}{ll} x + 3, & \text{if } x \geq -3 \\ -x - 3, & \text{if } x < -3 \\ \end{array}\right. \][/tex]
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