Question

select the correct answer. consider this absolute value function. $f(x)=|x + 3|$ if function $f$ is written as a piecewise function, which piece will it include? a. $-x + 3$, $x < -3$ b. $x + 3$, $x geq -3$ c. $x + 3$, $x geq 3$ d. $-x - 3$, $x < 3$

Explanation:

Step1: Find critical x-value

Set $x+3=0$, solve for $x$:
$x=-3$

Step2: Define absolute value rule

For $|A|$, $|A|=A$ if $A\geq0$, $|A|=-A$ if $A<0$.

Step3: Apply rule to $x+3$

When $x\geq-3$, $x+3\geq0$, so $f(x)=x+3$.
When $x<-3$, $x+3<0$, so $f(x)=-(x+3)=-x-3$.

Step4: Match to options

Compare with given choices.

Answer:

B. $x + 3,\ x \geq -3$