Choose the most correct answer. (A) \( f(x)=|x+7|=\left\{\begin{array}{ll}x+7 & \text { if } x \geq-7 \\ -x+7 & \text { if } x<-7\end{array}\right. \) (B) \( f(x)=|x+7|=\left\{\begin{array}{ll}x+7 & \text { if } x \geq 0 \\ -x+7 & \text { if } x<0\end{array}\right. \) (C) \( f(x)=|x+7|=\left\{\begin{array}{ll}x+7 & \text { if } x \geq 0 \\ -x-7 & \text { if } x<0\end{array}\right. \) (D) \( f(x)=|x+7|=\left\{\begin{array}{ll}x+7 & \text { if } x \geq-7 \\ -x-7 & \text { if } x<-7\end{array}\right. \)

To fully understand the function \( f(x) = |x+7| \), it's crucial to recognize that the absolute value function primarily concerns the expression inside it, \( x + 7 \). This function equals zero when \( x = -7 \). For \( x \geq -7 \), \( f(x) = x + 7 \) since the expression is non-negative. For \( x < -7 \), \( f(x) = -(x + 7) = -x - 7 \) because we flip the sign. Understanding these intervals ensures that we identify the correct piecewise function corresponding to the absolute value. Another interesting tidbit about absolute values is their real-world application. Absolute values are frequently used in scenarios that involve distances and differences. For instance, if you're measuring how far something is from a particular point, you might use an absolute value to ensure you're not getting a negative result, making it crucial in fields such as physics, engineering, and economics!