The functions f and g are defined as f(x)=x-2,g(x)=sqrt(x+3). a) Find the domain of f,g,f+g,f-g,fg,ff,(f)/(g), and (g)/(f). b) Find (f+g)(x),(f-g)(x),(fg)(x),(ff)(x),((f)/(g))(x), and ((g)/(f))(x).

Question

The functions  f and  g are defined as  f(x)=x-2,g(x)=sqrt(x+3). a) Find the domain of  f,g,f+g,f-g,fg,ff,(f)/(g), and  (g)/(f). b) Find  (f+g)(x),(f-g)(x),(fg)(x),(ff)(x),((f)/(g))(x), and  ((g)/(f))(x).

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Find Domain of f(x): To find the domain of f(x) = x - 2, we need to determine the set of all x-values for which f(x) is defined. Since f(x) is a polynomial function, it is defined for all real numbers.Find Domain of g(x): To find the domain of g(x) = sqrt(x + 3), we need to determine the set of all x-values for which g(x) is defined. The square root function is defined for non-negative arguments, so x + 3 must be greater than or equal to 0. Solving the inequality x + 3 >= 0 gives us x >= -3. Therefore, the domain of g(x) is all real numbers x such that x >= -3.Find Domain of f + g: To find the domain of f + g, we need to find the intersection of the domains of f and g. Since the domain of f is all real numbers and the domain of g is x >= -3, the domain of f + g is x >= -3.Find Domain of f - g: To find the domain of f - g, we again find the intersection of the domains of f and g. The domain of f - g is also x >= -3.Find Domain of fg: To find the domain of fg, we find the intersection of the domains of f and g. The domain of fg is x >= -3.Find Domain of ff: To find the domain of ff (f composed with f), we consider the domain of f, which is all real numbers. Since f is defined for all real numbers, the domain of ff is also all real numbers.Find Domain of (f)/(g): To find the domain of (f)/(g), we need to find the intersection of the domains of f and g, and exclude any x-values for which g(x) = 0. Since g(x) = sqrt(x + 3), g(x) is zero when x = -3. Therefore, the domain of (f)/(g) is x > -3.Find Domain of (g)/(f): To find the domain of (g)/(f), we need to find the intersection of the domains of f and g, and exclude any x-values for which f(x) = 0. Since f(x) = x - 2, f(x) is zero when x = 2. Therefore, the domain of (g)/(f) is x >= -3 and x ≠ 2.Calculate (f+g)(x): Now we will calculate (f+g)(x) = f(x) + g(x) = (x - 2) + sqrt(x + 3).Calculate (f-g)(x): Next, we calculate (f-g)(x) = f(x) - g(x) = (x - 2) - sqrt(x + 3).Calculate (fg)(x): Then, we calculate (fg)(x) = f(x) * g(x) = (x - 2) * sqrt(x + 3).Calculate (ff)(x): For (ff)(x), we calculate f(f(x)) = f(x - 2) = (x - 2) - 2 = x - 4.Calculate ((f)/(g))(x): To calculate ((f)/(g))(x), we find f(x) / g(x) = (x - 2) / sqrt(x + 3), for x > -3.Calculate ((g)/(f))(x): Finally, to calculate ((g)/(f))(x), we find g(x) / f(x) = sqrt(x + 3) / (x - 2), for x >= -3 and x ≠ 2.

The functions $f$ and $g$ are defined as $f(x)=x-2$ , $g(x)=\sqrt{x+3}$ . $\newline$ a) Find the domain of $\newline$ $f$ , $g$ , $f+g$ , $f-g$ , $fg$ , $f^2$ , $g$ $0$ , and $g$ $1$ . $\newline$ b) Find $\newline$ $g$ $2$ , $g$ $3$ , $g$ $4$ , $g$ $5$ , $g$ $6$ , and $g$ $7$ .

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The functions $f$ and $g$ are defined as $f(x)=x-2$ , $g(x)=\sqrt{x+3}$ . $\newline$ a) Find the domain of $\newline$ $f$ , $g$ , $f+g$ , $f-g$ , $fg$ , $f^2$ , $g$ $0$ , and $g$ $1$ . $\newline$ b) Find $\newline$ $g$ $2$ , $g$ $3$ , $g$ $4$ , $g$ $5$ , $g$ $6$ , and $g$ $7$ .