Answer :
To rewrite the absolute value function [tex]\( f(x) = |x+3| \)[/tex] as a piecewise function, you need to consider the behavior of the absolute value expression based on the value of [tex]\( x \)[/tex]. The absolute value function can be expressed as a piecewise function considering two cases:
1. When the expression inside the absolute value is non-negative, i.e., [tex]\( x + 3 \geq 0 \Rightarrow x \geq -3 \)[/tex].
2. When the expression inside the absolute value is negative, i.e., [tex]\( x + 3 < 0 \Rightarrow x < -3 \)[/tex].
For [tex]\( x \geq -3 \)[/tex], the absolute value function returns the expression itself:
[tex]\[ f(x) = x + 3 \][/tex]
For [tex]\( x < -3 \)[/tex], the absolute value function returns the negation of the expression:
[tex]\[ f(x) = -(x + 3) = -x - 3 \][/tex]
Putting it together, the piecewise function is:
[tex]\[ f(x) = \begin{cases} x + 3 & \text{if } x \geq -3 \\ -x - 3 & \text{if } x < -3 \end{cases} \][/tex]
So, the correct arrangement of the pieces is:
[tex]\[ f(x) = \begin{cases} x + 3 & \text{if } x \geq -3 \\ -x - 3 & \text{if } x < -3 \end{cases} \][/tex]
1. When the expression inside the absolute value is non-negative, i.e., [tex]\( x + 3 \geq 0 \Rightarrow x \geq -3 \)[/tex].
2. When the expression inside the absolute value is negative, i.e., [tex]\( x + 3 < 0 \Rightarrow x < -3 \)[/tex].
For [tex]\( x \geq -3 \)[/tex], the absolute value function returns the expression itself:
[tex]\[ f(x) = x + 3 \][/tex]
For [tex]\( x < -3 \)[/tex], the absolute value function returns the negation of the expression:
[tex]\[ f(x) = -(x + 3) = -x - 3 \][/tex]
Putting it together, the piecewise function is:
[tex]\[ f(x) = \begin{cases} x + 3 & \text{if } x \geq -3 \\ -x - 3 & \text{if } x < -3 \end{cases} \][/tex]
So, the correct arrangement of the pieces is:
[tex]\[ f(x) = \begin{cases} x + 3 & \text{if } x \geq -3 \\ -x - 3 & \text{if } x < -3 \end{cases} \][/tex]
