The function \( p \) is defined as \( p(x)=\left\{\begin{array}{ll}2 x+1 & \text { if } x>1 \\ -x^{2}+1 & \text { if }-1

Ask by Murray Burns. in South Africa

Feb 26,2025

To find the domain of the function \( p(x) \), we need to analyze the piecewise definition. The first part, \( 2x + 1 \), is defined for \( x > 1 \). The second part, \( -x^2 + 1 \), is defined for \( -1 < x < 1 \). Combining these intervals, we see that \( p(x) \) is defined for \( x \) values in \( (-1, 1) \) and \( (1, \infty) \). However, \( p(x) \) is not defined at \( x = 1 \) as neither piece applies. Therefore, the domain of function \( p \) is: \( \mathrm{D} p=(-1, 1) \cup (1, \infty) \). The correct answer is (E). Now for some fun facts! The piecewise function is a great way to explore mathematical behavior across different intervals. It's like a mathematical buffet where each dish (or equation) serves a different flavor depending on the x-value! Understanding domains and ranges can help in real-world applications too! In economics, for example, a profit function may only apply in certain conditions. By identifying these intervals where the function works best, businesses can make smarter decisions!