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The problem asks for:

1. The **domain**, **range**, and **graph** of the function \( f(x) = \sqrt{x^2 - 1} \) and \( g(x) = \frac{2}{x} \).
2. The **domain** and **composition** of \( f(g(x)) \).

Let's break it down:

### a) Domain, Range, and Graph of \( f(x) = \sqrt{x^2 - 1} \)
#### Domain:
For the square root function to be real, the expression inside the square root must be non-negative:
\[
x^2 - 1 \geq 0
\]
Solving this inequality:
\[
x^2 \geq 1 \quad \Rightarrow \quad |x| \geq 1
\]
So, the domain of \( f(x) \) is:
\[
x \in (-\infty, -1] \cup [1, \infty)
\]

#### Range:
Since \( f(x) = \sqrt{x^2 - 1} \), the smallest value \( x^2 - 1 \) can take is \( 0 \), and there is no upper bound. Therefore, the range is:
\[
[0, \infty)
\]

#### Graph:
The graph of \( f(x) = \sqrt{x^2 - 1} \) resembles a curve starting from \( (1, 0) \) and \( (-1, 0) \), and increases as \( |x| \) increases.

---

### b) Domain and Composition of \( f(g(x)) \)

#### Domain of \( g(x) = \frac{2}{x} \):
The function \( g(x) \) is undefined at \( x = 0 \), so the domain is:
\[
x \in (-\infty, 0) \cup (0, \infty)
\]

#### Composition \( f(g(x)) \):
For \( f(g(x)) = f\left( \frac{2}{x} \right) \), we need to ensure that the argument of \( f(x) \), which is \( \frac{2}{x} \), satisfies the domain condition of \( f(x) \).

We require:
\[
\left( \frac{2}{x} \right)^2 - 1 \geq 0
\]
Simplifying:
\[
\frac{4}{x^2} - 1 \geq 0
\quad \Rightarrow \quad \frac{4 - x^2}{x^2} \geq 0
\]
This inequality holds when:
\[
x^2 \geq 4 \quad \Rightarrow \quad |x| \geq 2
\]
Thus, the domain of \( f(g(x)) \) is:
\[
x \in (-\infty, -2] \cup [2, \infty)
\]

Would you like more details on any part? Here are a few questions that could expand this:

1. How do you find the derivative of \( f(x) \)?
2. What is the graph of \( g(x) \)?
3. Can you express \( f(g(x)) \) explicitly?
4. How do you solve inequalities involving square roots?
5. What are the applications of function composition in calculus?

**Tip**: Always double-check domains when dealing with square roots or rational functions to ensure valid inputs.

Diketahui f(x) = √(x^2 - 1) dan g(x) = 2/x. a) Domain, Range, dan Grafik f(x) b) Domain dan Komposisi f(g(x))

Find the domain, range, and graph of f(x) = √(x^2 - 1) and g(x) = 2/x. Learn how to determine the domain and composition of functions in this detailed solution for Grades 10-12.

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