Which of the following pairs are identical?

Options

A. `f(x)=sqrt(x^(2)), g(x)=(sqrt(x))^(2)`
B. `f(x)=(1)/(sqrt(x^(2))) , g(x)= (x)/(x^(2))`
C. `f(x) =log(x-1) +log(x-2), g(x)=log(x-1) (x-2)`
D. None of these

Answer

To determine which of the given pairs of functions are identical, we will analyze each pair step by step.

### Step 1: Analyze the first pair of functions
Let’s denote the first pair as:
- \( f(x) = \sqrt{x^2} \)
- \( g(x) = \sqrt{x^2} \)

**Domain of \( f(x) \)**:
The function \( f(x) = \sqrt{x^2} \) is defined for all real numbers \( x \). This is because the square root of a square is always non-negative. Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (-\infty, \infty) \]

**Domain of \( g(x) \)**:
Similarly, \( g(x) = \sqrt{x^2} \) is also defined for all real numbers \( x \). Therefore, the domain of \( g(x) \) is:
\[ \text{Domain of } g(x) = (-\infty, \infty) \]

**Conclusion for the first pair**:
Since both functions have the same domain, they are identical.

### Step 2: Analyze the second pair of functions
Let’s denote the second pair as:
- \( f(x) = \frac{1}{x} \)
- \( g(x) = \frac{1}{x} \)

**Domain of \( f(x) \)**:
The function \( f(x) = \frac{1}{x} \) is defined for all real numbers except \( x = 0 \). Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (-\infty, 0) \cup (0, \infty) \]

**Domain of \( g(x) \)**:
Similarly, \( g(x) = \frac{1}{x} \) has the same domain:
\[ \text{Domain of } g(x) = (-\infty, 0) \cup (0, \infty) \]

**Conclusion for the second pair**:
Since both functions have the same domain, they are identical.

### Step 3: Analyze the third pair of functions
Let’s denote the third pair as:
- \( f(x) = \log(x-1) + \log(x-2) \)
- \( g(x) = \log((x-1)(x-2)) \)

**Domain of \( f(x) \)**:
The function \( f(x) \) is defined when both \( x-1 > 0 \) and \( x-2 > 0 \). This means:
- \( x > 1 \)
- \( x > 2 \)

Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (2, \infty) \]

**Domain of \( g(x) \)**:
The function \( g(x) \) is defined when \( (x-1)(x-2) > 0 \). This occurs when:
- \( x < 1 \) or \( x > 2 \)

Thus, the domain of \( g(x) \) is:
\[ \text{Domain of } g(x) = (-\infty, 1) \cup (2, \infty) \]

**Conclusion for the third pair**:
Since the domains of \( f(x) \) and \( g(x) \) are different, they are not identical.

### Final Conclusion
The identical pairs are:
- Pair 1: \( f(x) = \sqrt{x^2} \) and \( g(x) = \sqrt{x^2} \)
- Pair 2: \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x} \)