For the piecewise-defined function below, state the equivalent absolute value function. $$ f(x)=\left\{\begin{array}{ll}-3 x+21 & \text { if } x \leq 7 \ 3 x-21 & \text { if } x7\end{array}\right. $$ The equivalent absolute value function is $$ f(x)=\square $$ 7

Question

For the piecewise-defined function below, state the equivalent absolute value function. $$ f(x)=\left\{\begin{array}{ll}-3 x+21 & \text { if } x \leq 7 \ 3 x-21 & \text { if } x>7\end{array}\right. $$ The equivalent absolute value function is $$ f(x)=\square $$ 7

UpStudy AI · Accepted Answer

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

1. For $$ x \leq 7 $$:
   $$
   f(x) = -3x + 21
   $$

2. For $$ x > 7 $$:
   $$
   f(x) = 3x - 21
   $$

Next, we can find a way to express both cases in terms of an absolute value function.

Notice that both expressions can be rewritten in terms of the point $$ x = 7 $$:

- For $$ x \leq 7 $$:
  $$
  f(x) = -3(x - 7) \quad \text{(since } -3x + 21 = -3(x - 7) \text{)}
  $$

- For $$ x > 7 $$:
  $$
  f(x) = 3(x - 7) \quad \text{(since } 3x - 21 = 3(x - 7) \text{)}
  $$

Now, we can combine these into a single absolute value function. The absolute value function that captures both cases is:
$$
f(x) = 3|x - 7|
$$

Thus, the equivalent absolute value function is:
$$
f(x) = 3|x - 7|
$$
The equivalent absolute value function is $$ f(x) = 3|x - 7| $$.

For the piecewise-defined function below, state the equivalent absolute value function. \[ f(x)=\left\{\begin{array}{ll}-3 x+21 & \text { if } x \leq 7 \\ 3 x-21 & \text { if } x>7\end{array}\right. \] The equivalent absolute value function is \( f(x)=\square \) 7

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