To find \((f \cdot g + h)(x)\), we will first determine \(f(x)\), \(g(x)\), and \(h(x)\), and then combine them accordingly.
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We have:
\[ f(x) = x^2 + 1 \]
\[ g(x) = 3x - 1 \]
\[ h(x) = |x| + 2 \]
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Next, we calculate \(f \cdot g\):
\[ f \cdot g = f(x) \cdot g(x) = (x^2 + 1)(3x - 1) \]
To multiply these two functions, we distribute:
\[ (x^2 + 1)(3x - 1) = x^2 \cdot 3x + x^2 \cdot (-1) + 1 \cdot 3x + 1 \cdot (-1) \]
This simplifies to:
\[ 3x^3 - x^2 + 3x - 1 \]
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Now, we add \(h(x)\) to \(f \cdot g\):
\[ (f \cdot g + h)(x) = (3x^3 - x^2 + 3x - 1) + (|x| + 2) \]
Combining terms gives us:
\[ 3x^3 - x^2 + 3x - 1 + |x| + 2 \]
Which simplifies to:
\[ 3x^3 - x^2 + 3x + |x| + 1 \]
So, the final expression for \((f \cdot g + h)(x)\) is:
\[ 3x^3 - x^2 + 3x + |x| + 1 \]
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