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oxed{(0, 4)}
Question: If $ x + rac{1}{2} = 3 $, what is $ 4x + 2 $?
Solution: Solve $ x = 3 - rac{1}{2} = rac{5}{2} $. Substitute into $ 4x + 2 $: $ 4 imes rac{5}{2} + 2 = 10 + 2 = 12 $.
oxed{12}Question: If the revenue of a startup grows exponentially as $ R(t) = 1000 \cdot e^{kt} $ and doubles in 4 years, find the value of $ k $.
Solution: Given $ R(4) = 2R(0) $, substitute into the equation: $ 1000e^{4k} = 2 \cdot 1000 $. Simplify to $ e^{4k} = 2 $. Take the natural logarithm: $ 4k = \ln(2) $, so $ k = rac{\ln(2)}{4} $. Thus, $ oxed{\dfrac{\ln(2)}{4}} $.
Question: The self-healing efficiency of a polymer is modeled by $ rac{x^2 - 9}{x - 3} = 5 $. Solve for $ x $.
Solution: Simplify the left side: $ rac{(x-3)(x+3)}{x-3} = x + 3 $, provided $ x
eq 3 $. Set $ x + 3 = 5 $, so $ x = 2 $. Verify $ x = 2 $ does not violate the restriction. Final answer: $ oxed{2} $.
Question: A quantum sensing function satisfies $ f(x) = 2x + 1 $ and $ g(x) = x^2 - 4 $. Compute $ f(g(3)) $.
Solution: First compute $ g(3) = 3^2 - 4 = 5 $. Then $ f(g(3)) = f(5) = 2(5) + 1 = 11 $. Final answer: $ oxed{11} $.