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\( \left. \begin{array} { l } { f ( x ) = ( - x ) ^ { 3 } - x \cdot - x + 11 } \\ { a \cdot f ( 3 ) } \\ { b \cdot f ( - 3 ) } \end{array} \right. \)

Ask by Osborne Johnson. in the United States
Feb 22,2025

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a·f(3) = –7a b·f(–3) = 47b

Solución

We start with the function   f(x) = (–x)³ – x·(–x) + 11. Step 1. Simplify f(x):  • (–x)³ = –x³                   [since (–1)³ = –1]  • x·(–x) = –x², so –[x·(–x)] = +x². Thus, f(x) = –x³ + x² + 11. Step 2. Evaluate f(3):  • f(3) = –(3)³ + (3)² + 11     = –27 + 9 + 11     = –27 + 20     = –7. So, a·f(3) becomes a·(–7) = –7a. Step 3. Evaluate f(–3):  • f(–3) = –(–3)³ + (–3)² + 11     First, compute (–3)³ = –27, so –(–27) = 27.     Next, (–3)² = 9.     Then, f(–3) = 27 + 9 + 11         = 47. So, b·f(–3) becomes b·47 = 47b. Final Answers:   a·f(3) = –7a    and  b·f(–3) = 47b.

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Let’s dive into the function \( f(x) = (-x)^3 - x \cdot -x + 11 \). If we first simplify it, we see that \( (-x)^3 = -x^3 \) and \( -x \cdot -x = x^2 \). So \( f(x) \) becomes \( f(x) = -x^3 + x^2 + 11 \). Now, to find \( f(3) \): \[ f(3) = -3^3 + 3^2 + 11 = -27 + 9 + 11 = -7. \] Next, for \( f(-3) \): \[ f(-3) = -(-3)^3 + (-3)^2 + 11 = 27 + 9 + 11 = 47. \] So \( a \cdot f(3) = a \cdot (-7) \) and \( b \cdot f(-3) = b \cdot 47 \). You can play with values of \( a \) and \( b \) to get interesting results!

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