Which shows \( f(x)=|x-24| \) expressed as a piecewise-defined function? \[ f(x)=\left\{\begin{array}{l}x-24, x \geq 24 \\ -x+24, x<24\end{array}\right. \] \[ f(x)=\left\{\begin{array}{l}x-24, x \geq-24 \\ -x+24, x<-24\end{array}\right. \] \( f(x)=\left\{\begin{array}{l}x-24, x \geq 0 \\ -x+24, x<0\end{array}\right. \) \( f(x)=\left\{\begin{array}{l}x-24, x \geq 1 / 24 \\ -x+24, x<1 / 24\end{array}\right. \)

To express the function \( f(x) = |x - 24| \) as a piecewise-defined function, we need to consider the definition of the absolute value function. The absolute value \( |a| \) can be defined as: \[ |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} \] In our case, we have \( a = x - 24 \). Therefore, we need to determine when \( x - 24 \) is non-negative and when it is negative. 1. **When is \( x - 24 \geq 0 \)?** - This occurs when \( x \geq 24 \). 2. **When is \( x - 24 < 0 \)?** - This occurs when \( x < 24 \). Using this information, we can write the piecewise definition of \( f(x) \): \[ f(x) = \begin{cases} x - 24 & \text{if } x \geq 24 \\ -(x - 24) & \text{if } x < 24 \end{cases} \] Simplifying the second case: \[ -(x - 24) = -x + 24 \] Thus, the piecewise-defined function becomes: \[ f(x) = \begin{cases} x - 24 & \text{if } x \geq 24 \\ -x + 24 & \text{if } x < 24 \end{cases} \] Now, let's compare this with the options provided: 1. \( f(x)=\left\{\begin{array}{l}x-24, x \geq 24 \\ -x+24, x<24\end{array}\right. \) (This matches our derived function.) 2. \( f(x)=\left\{\begin{array}{l}x-24, x \geq-24 \\ -x+24, x<-24\end{array}\right. \) (Incorrect, as it does not match the condition \( x = 24 \).) 3. \( f(x)=\left\{\begin{array}{l}x-24, x \geq 0 \\ -x+24, x<0\end{array}\right. \) (Incorrect, as it does not match the condition \( x = 24 \).) 4. \( f(x)=\left\{\begin{array}{l}x-24, x \geq 1 / 24 \\ -x+24, x<1 / 24\end{array}\right. \) (Incorrect, as it does not match the condition \( x = 24 \).) The correct piecewise-defined function for \( f(x) = |x - 24| \) is: \[ f(x)=\left\{\begin{array}{l}x-24, x \geq 24 \\ -x+24, x<24\end{array}\right. \] Thus, the first option is the correct representation of \( f(x) \).