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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

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The domain of function \( p \) is \( (-1, 1) \cup (1, \infty) \), which corresponds to option (E).

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We need to find all values of x for which p(x) is defined. The function p is defined by two cases: 1. p(x) = 2x + 1 for x > 1. 2. p(x) = –x² + 1 for –1 < x < 1. Notice: • For x > 1, p(x) is defined by the first case. • For x between –1 and 1 (but not including –1 or 1), p(x) is defined by the second case. • There is no rule given for x ≤ –1, and there is no rule for x = 1. Thus, the definition of p covers x in the intervals (–1, 1) and (1, ∞). Combining these, we get:   Domain of p = (–1, 1) ∪ (1, ∞). Comparing this with the options provided, the correct answer is Option (E).

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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!

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