The task you've given involves several steps, so let's break it down:

1. Understanding the functions: We have two functions, f(x) = 1/x and g(x) = x/(x-1).

2. Calculating the functional values:

(i) f+g(x) = f(x) + g(x) = 1/x + x/(x-1). To simplify this, we need to find a common denominator, which is x(x-1). So, f+g(x) = (1*(x-1) + x*x) / x*(x-1) = (x^2 - 1) / x*(x-1).

(ii) (fg)(x) = f(x) * g(x) = (1/x) * (x/(x-1)) = 1/(x-1).

(iii) fog(x) = f(g(x)) = 1/g(x) = 1/(x/(x-1)) = x-1.

(iv) gof(x) = g(f(x)) = f(1/x) = (1/x) / ((1/x) - 1) = 1 / (1 - x).

3. Checking if fg, fog, and gof are equal: From the above calculations, we can see that (fg)(x) = 1/(x-1), fog(x) = x-1, and gof(x) = 1 / (1 - x). So, fg, fog, and gof are not equal.

4. Finding the domain and range:

(i) For f+g(x), the domain is all real numbers except 0 and 1, because those make the denominator zero. The range is all real numbers, because the numerator can be any real number.

(ii) For (fg)(x), the domain is all real numbers except 1, because that makes the denominator zero. The range is all real numbers except 0, because the function never equals 0.

(iii) For fog(x), the domain is all real numbers except 1, because that makes the denominator of g(x) zero. The range is all real numbers except 1, because the function never equals 1.

(iv) For gof(x), the domain is all real numbers except 0 and 1, because those make the denominator zero. The range is all real numbers, because the numerator can be any real number.

Remember, the domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

Question

The task you've given involves several steps, so let's break it down:

1. Understanding the functions: We have two functions, f(x) = 1/x and g(x) = x/(x-1).

2. Calculating the functional values:

(i) f+g(x) = f(x) + g(x) = 1/x + x/(x-1). To simplify this, we need to find a common denominator, which is x(x-1). So, f+g(x) = (1*(x-1) + x*x) / x*(x-1) = (x^2 - 1) / x*(x-1).

(ii) (fg)(x) = f(x) * g(x) = (1/x) * (x/(x-1)) = 1/(x-1).

(iii) fog(x) = f(g(x)) = 1/g(x) = 1/(x/(x-1)) = x-1.

(iv) gof(x) = g(f(x)) = f(1/x) = (1/x) / ((1/x) - 1) = 1 / (1 - x).

3. Checking if fg, fog, and gof are equal: From the above calculations, we can see that (fg)(x) = 1/(x-1), fog(x) = x-1, and gof(x) = 1 / (1 - x). So, fg, fog, and gof are not equal.

4. Finding the domain and range:

(i) For f+g(x), the domain is all real numbers except 0 and 1, because those make the denominator zero. The range is all real numbers, because the numerator can be any real number.

(ii) For (fg)(x), the domain is all real numbers except 1, because that makes the denominator zero. The range is all real numbers except 0, because the function never equals 0.

(iii) For fog(x), the domain is all real numbers except 1, because that makes the denominator of g(x) zero. The range is all real numbers except 1, because the function never equals 1.

(iv) For gof(x), the domain is all real numbers except 0 and 1, because those make the denominator zero. The range is all real numbers, because the numerator can be any real number.

Remember, the domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

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