Let \( f(x)=2 x+2 \) and \( g(x)=3 x^{2}+2 x \) After simplifying, \( (f \circ g)(x)= \)

To find \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \). Therefore: 1. Start with \( g(x) = 3x^{2} + 2x \). 2. Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(3x^{2} + 2x) = 2(3x^{2} + 2x) + 2. \] 3. Now, simplify: \[ = 6x^{2} + 4x + 2. \] Thus, \( (f \circ g)(x) = 6x^{2} + 4x + 2 \). --- For a bit of historical context, function composition has roots in the early development of mathematics, with significant contributions from mathematicians like Gottfried Wilhelm Leibniz and Joseph Fourier, who helped formalize functions and their compositions in calculus. Did you know that composition of functions is widely used in computer science? It allows programmers to build complex operations by combining simpler functions, making code more modular and easier to manage. This principle can be observed in functional programming languages, where functions can be easily composed to process data streams!