To find \((f \circ g)(x)\), we first need to understand what it means to compose two functions. The expression \((f \circ g)(x)\) means to apply the function \(g\) first, and then apply \(f\) to the result of \(g\).
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Given Functions:
- \(f(x) = 0.04x\)
- \(g(x) = x - 4,000\)
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Composition: \[ (f \circ g)(x) = f(g(x)) = f(x - 4,000) \] Now, substitute \(g(x)\) into \(f(x)\): \[ f(x - 4,000) = 0.04(x - 4,000) \] Simplifying this: \[ f(x - 4,000) = 0.04x - 0.04 \times 4,000 = 0.04x - 160 \] Therefore: \[ (f \circ g)(x) = 0.04x - 160 \]
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Meaning: \((f \circ g)(x) = 0.04x - 160\) represents the amount Sally earns in commission after accounting for the threshold of $4,000 in sales. The term \(-160\) indicates that she starts earning commission only after reaching $4,000 in sales, as she doesn't earn commission on the first $4,000.
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Domain: Since \(g(x) = x - 4,000\) (i.e., Sally earns commission only on sales exceeding $4,000), for \(g(x)\) to be valid and represent applicable sales, \(x\) must be greater than or equal to $4,000. Thus, the minimum \(x\) value for which commission can be calculated is $4,000. Therefore, \(x \geq 4,000\).
Overall, the correct interpretation is: \[ (f \circ g)(x) = 0.04x - 160 \quad \text{(represents amount earned in commission)} \] Domain: \(x \geq 4,000\).
The correct selection would therefore match: \((f \circ g)(x) = 0.04x - 160 \text{ represents the amount Sally earns in commission. The domain is } x \geq 4,000.\)
1 answer