5. At \( x=3 \), the function given by \( f(x)=\left\{\begin{array}{ll}x^{2}, & x<3 \\ 6 x-9, & x \geq 3\end{array}\right. \) is

To evaluate the function \( f(x) \) at \( x=3 \), we need to look at the piece of the function that applies when \( x \) is equal to or greater than 3. Here, that piece is \( 6x - 9 \). Substituting \( x = 3 \) into that expression, we get: \[ f(3) = 6(3) - 9 = 18 - 9 = 9. \] Thus, at \( x=3 \), the function evaluates to \( f(3) = 9 \). Now, let’s talk about some interesting aspects! Did you know that piecewise functions can model real-world situations? For instance, they can represent scenarios where different rules apply under different conditions, like determining shipping costs based on the weight of a package! Additionally, when working with piecewise functions, it’s easy to overlook the boundaries. A common mistake is to neglect checking the correct piece at the boundaries—like when \( x = 3 \) in this case. Always remember to read the conditions carefully!