let f(x) = cos(5x) and g(x) = x/(x + 4). determine the following combined functions and their respective…

1. ((f + g)(x)) and its domain:
   • Step - 1: Find the combined function ((f + g)(x))
     • By the definition of the sum of two functions ((f + g)(x)=f(x)+g(x)). Given (f(x)=\cos(5x)) and (g(x)=\frac{x}{x + 4}), then ((f + g)(x)=\cos(5x)+\frac{x}{x + 4}).
   • Step - 2: Find the domain of ((f + g)(x))
     • The domain of (y = \cos(5x)) is all real numbers, i.e., ((-\infty,\infty)). The domain of (y=\frac{x}{x + 4}) is all real - numbers except the value of (x) that makes the denominator zero. Set (x + 4=0), then (x=-4).
     • The domain of ((f + g)(x)) is the intersection of the domains of (f(x)) and (g(x)). So the domain is ((-\infty,-4)\cup(-4,\infty)).
2. ((f - g)(x)) and its domain:
   • Step - 1: Find the combined function ((f - g)(x))
     • By the definition of the difference of two functions ((f - g)(x)=f(x)-g(x)). So ((f - g)(x)=\cos(5x)-\frac{x}{x + 4}).
   • Step - 2: Find the domain of ((f - g)(x))
     • Similar to the sum case, the domain of (\cos(5x)) is ((-\infty,\infty)) and the domain of (\frac{x}{x + 4}) is all real numbers except (x=-4). The domain of ((f - g)(x)) is ((-\infty,-4)\cup(-4,\infty)).
3. ((f\cdot g)(x)) and its domain:
   • Step - 1: Find the combined function ((f\cdot g)(x))
     • By the definition of the product of two functions ((f\cdot g)(x)=f(x)\cdot g(x)). So ((f\cdot g)(x)=\cos(5x)\cdot\frac{x}{x + 4}=\frac{x\cos(5x)}{x + 4}).
   • Step - 2: Find the domain of ((f\cdot g)(x))
     • The domain of (\cos(5x)) is ((-\infty,\infty)) and the domain of (\frac{x}{x + 4}) is all real numbers except (x=-4). The domain of ((f\cdot g)(x)) is ((-\infty,-4)\cup(-4,\infty)).
4. ((\frac{f}{g})(x)) and its domain:
   • Step - 1: Find the combined function ((\frac{f}{g})(x))
     • By the definition of the quotient of two functions ((\frac{f}{g})(x)=\frac{f(x)}{g(x)}), where (g(x)\neq0). So ((\frac{f}{g})(x)=\frac{\cos(5x)}{\frac{x}{x + 4}}=\frac{(x + 4)\cos(5x)}{x}), with (x\neq0) and (x\neq - 4).
   • Step - 2: Find the domain of ((\frac{f}{g})(x))
     • The domain of (f(x)=\cos(5x)) is ((-\infty,\infty)), the domain of (g(x)=\frac{x}{x + 4}) is (x\neq - 4). For the quotient (\frac{f(x)}{g(x)}), we also need (g(x)\neq0), i.e., (x\neq0). So the domain is ((-\infty,-4)\cup(-4,0)\cup(0,\infty)).

Answer:

• ((f + g)(x)=\cos(5x)+\frac{x}{x + 4}), domain: ((-\infty,-4)\cup(-4,\infty))
• ((f - g)(x)=\cos(5x)-\frac{x}{x + 4}), domain: ((-\infty,-4)\cup(-4,\infty))
• ((f\cdot g)(x)=\frac{x\cos(5x)}{x + 4}), domain: ((-\infty,-4)\cup(-4,\infty))
• ((\frac{f}{g})(x)=\frac{(x + 4)\cos(5x)}{x}), domain: ((-\infty,-4)\cup(-4,0)\cup(0,\infty))