3.4E: Composition of Functions (Exercises)

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For the following exercises, find \((f \circ g)(x)\) and \((g \circ f)(x)\) for each pair of functions.

34. \(f(x)=4-x, \quad g(x)=-4 x\)
35. \(f(x)=3 x+2, \quad g(x)=5-6 x\)
36. \(f(x)=x^{2}+2 x, \quad g(x)=5 x+1\)
37. \(f(x)=\sqrt{x+2}, \quad g(x)=\frac{1}{x}\)
38. \(f(x)=\frac{x+3}{2}, \quad g(x)=\sqrt{1-x}\)

For the following exercises, find \((f \circ g)\) and the domain for \((f \circ g)(x)\) for each pair of functions.

39. \(f(x)=\frac{x+1}{x+4}, \quad g(x)=\frac{1}{x}\)
40. \(f(x)=\frac{1}{x+3}, \quad g(x)=\frac{1}{x-9}\)
41. \(f(x)=\frac{1}{x}, \quad g(x)=\sqrt{x}\)
42. \(f(x)=\frac{1}{x^{2}-1}, \quad g(x)=\sqrt{x+1}\)

For the following exercises, express each function \(H\) as a composition of two functions \(f\) and \(g\) where \(H(x)=(f \circ g)(x)\).

43. \(H(x)=\sqrt{\frac{2 x-1}{3 x+4}}\)
44. \(H(x)=\frac{1}{\left(3 x^{2}-4\right)^{-3}}\)