For the piecewise-defined function below, state the equivalent absolute value function. $$ f(x)=\left\{\begin{array}{cc}-2 x+12 & \text { if } x \leq 6 \ 2 x-12 & \text { if } x6\end{array}\right. $$ The equivalent absolute value function is $$ f(x)=\square $$

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For the piecewise-defined function below, state the equivalent absolute value function. $$ f(x)=\left\{\begin{array}{cc}-2 x+12 & \text { if } x \leq 6 \ 2 x-12 & \text { if } x>6\end{array}\right. $$ The equivalent absolute value function is $$ f(x)=\square $$

UpStudy AI · Accepted Answer

To express the given piecewise-defined function as an absolute value function, we need to analyze the two cases.

1. For $$ x \leq 6 $$:
   $$
   f(x) = -2x + 12
   $$
   This can be rewritten as:
   $$
   f(x) = -2(x - 6)
   $$

2. For $$ x > 6 $$:
   $$
   f(x) = 2x - 12
   $$
   This can be rewritten as:
   $$
   f(x) = 2(x - 6)
   $$

Now, we can combine these two cases into a single absolute value function. The function changes its behavior at $$ x = 6 $$, which suggests that we can express it in terms of $$ |x - 6| $$.

The absolute value function that captures both cases is:
$$
f(x) = 2|x - 6|
$$

Thus, the equivalent absolute value function is:
$$
f(x) = 2|x - 6|
$$
The equivalent absolute value function is $$ f(x) = 2|x - 6| $$.

For the piecewise-defined function below, state the equivalent absolute value function. \[ f(x)=\left\{\begin{array}{cc}-2 x+12 & \text { if } x \leq 6 \\ 2 x-12 & \text { if } x>6\end{array}\right. \] The equivalent absolute value function is \( f(x)=\square \)

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