1. The problem is to write the piecewise function for $f(|x|)$ given a function $f(x)$. 2. Recall that the absolute value function $|x|$ is defined as: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ 3. To express $f(|x|)$ as a piecewise function, we substitute $|x|$ with its definition: $$f(|x|) = \begin{cases} f(x) & \text{if } x \geq 0 \\ f(-x) & \text{if } x < 0 \end{cases}$$ 4. This means for non-negative $x$, $f(|x|)$ equals $f(x)$, and for negative $x$, $f(|x|)$ equals $f(-x)$. 5. This piecewise form is useful because it allows us to analyze $f(|x|)$ by considering the behavior of $f$ on positive inputs and reflecting it for negative inputs. Final answer: $$f(|x|) = \begin{cases} f(x) & x \geq 0 \\ f(-x) & x < 0 \end{cases}$$