12 The function \( f \) is defined for all real \( x \) by \( f(x)=\left\{\begin{array}{l}\frac{1}{x} \text { for } x>0 \\ x \mid \text { for } x \leq 0\end{array}\right. \) Find the set of values of \( x \) for which \( f(x) \leq 4 \)

To find the set of values of \( x \) for which \( f(x) \leq 4 \), we analyze the function piece by piece. 1. For \( x > 0 \): Here, \( f(x) = \frac{1}{x} \). We want to solve the inequality: \[ \frac{1}{x} \leq 4 \] Multiplying both sides by \( x \) (which is positive, so the inequality sign stays the same), we get: \[ 1 \leq 4x \] Dividing both sides by 4, we have: \[ \frac{1}{4} \leq x \] So for \( x > 0 \), the solution set is: \[ x \in \left[\frac{1}{4}, \infty\right) \] 2. For \( x \leq 0 \): Here, \( f(x) = x \). We want to solve the inequality: \[ x \leq 4 \] Since \( x \) is already restricted to values less than or equal to 0, all values in this region satisfy the inequality: \[ x \in (-\infty, 0] \] Finally, we combine both results: - For \( x > 0 \), the solution is \( \left[\frac{1}{4}, \infty\right) \). - For \( x \leq 0 \), the solution is \( (-\infty, 0] \). Therefore, the set of values of \( x \) for which \( f(x) \leq 4 \) is: \[ x \in \left(-\infty, 0\right] \cup \left[\frac{1}{4}, \infty\right) \]