(ii)

The composition of two functions, (f∘g)(x), is the function obtained by applying f to the result of applying g to x. In other words, (f∘g)(x) = f(g(x)).

Given f(x) = √x - 1 and g(x) = 1/(x - 2), we want to find (f∘g)(x).

So, (f∘g)(x) = f(g(x)) = f(1/(x - 2)).

Substitute g(x) into f, we get:

f(g(x)) = √(1/(x - 2)) - 1.

This is the simplified expression for (f∘g)(x).

Now, let's find the domain of (f∘g)(x).

The domain of (f∘g)(x) is the set of all x such that g(x) is in the domain of f and x is in the domain of g.

From part (i), we know that the domain of f is [1, ∞) and the domain of g is (-∞, 2) U (2, ∞).

Therefore, we need to find all x such that 1/(x - 2) ≥ 1 (because the domain of f is [1, ∞)) and x ≠ 2 (because 2 is not in the domain of g).

Solving the inequality 1/(x - 2) ≥ 1, we get x ≤ 3.

So, the domain of (f∘g)(x) is (-∞, 2) U (2, 3].

Question

(ii)

The composition of two functions, (f∘g)(x), is the function obtained by applying f to the result of applying g to x. In other words, (f∘g)(x) = f(g(x)).

Given f(x) = √x - 1 and g(x) = 1/(x - 2), we want to find (f∘g)(x).

So, (f∘g)(x) = f(g(x)) = f(1/(x - 2)).

Substitute g(x) into f, we get:

f(g(x)) = √(1/(x - 2)) - 1.

This is the simplified expression for (f∘g)(x).

Now, let's find the domain of (f∘g)(x).

The domain of (f∘g)(x) is the set of all x such that g(x) is in the domain of f and x is in the domain of g.

From part (i), we know that the domain of f is [1, ∞) and the domain of g is (-∞, 2) U (2, ∞).

Therefore, we need to find all x such that 1/(x - 2) ≥ 1 (because the domain of f is [1, ∞)) and x ≠ 2 (because 2 is not in the domain of g).

Solving the inequality 1/(x - 2) ≥ 1, we get x ≤ 3.

So, the domain of (f∘g)(x) is (-∞, 2) U (2, 3].

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