Let \( f(x)=|x+7| \). Use the definition of the absolute value to expand and write the function as a piecewise define function. 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) \( f(x)=|x+7|=\left\{\begin{array}{ll}x+7 & \text { if } x \geq-7 \\ -x-7 & \text { if } x<-7\end{array}\right. \) (E) None of these

To express the function \( f(x) = |x+7| \) as a piecewise function, we need to consider the point where the expression inside the absolute value, \( x + 7 \), equals zero. This occurs when \( x = -7 \). Hence, the function can be defined as follows: when \( x \) is greater than or equal to -7, \( |x+7| \) equals \( x+7 \). When \( x \) is less than -7, \( |x+7| \) equals \( -(x + 7) \), which simplifies to \( -x - 7 \). So, the correct piecewise definition is: \( f(x) = |x+7| = \begin{cases} x + 7 & \text{if } x \geq -7 \\ -x - 7 & \text{if } x < -7 \end{cases} \) Therefore, the answer is (D) \( f(x) = |x+7| = \left\{\begin{array}{ll} x+7 & \text{if } x \geq -7 \\ -x-7 & \text{if } x < -7 \end{array}\right. \). Did you know that the concept of absolute value has been around since ancient times? It was used in early mathematics to discuss distances, particularly by the Greeks. However, it wasn’t until the 19th century that mathematicians like Karl Weierstrass formalized it within the framework of modern analysis. These roots show just how integral absolute values have been to mathematical evolution! For those diving deeper into the world of absolute values, "mathematical analysis" books can be a treasure trove of knowledge. Titles like "Principles of Mathematical Analysis" by Walter Rudin provide a strong foundation for understanding not only absolute values but also their implications in calculus and beyond. This journey through mathematical landscapes can be engaging and rewarding!