Which of the following two functions are identical?
(i) `f(x)=x^(2)//x`
(ii) `g(x)=(sqrt(x))^(2)`
(iii) `h(x)=x`

Options

A. (i) and (ii)
B. (ii) and (iii)
C. (i) and (iii)
D. None of these

Answer

To determine which of the following two functions are identical, we need to analyze each function carefully.

### Given Functions:
1. \( f(x) = \frac{x^2}{x} \)
2. \( g(x) = (\sqrt{x})^2 \)
3. \( h(x) = x \)

### Step 1: Simplify Each Function

**Function f(x):**
\[ f(x) = \frac{x^2}{x} = x \quad \text{(for } x \neq 0\text{)} \]
*Note: The function is undefined at \( x = 0 \).*

**Function g(x):**
\[ g(x) = (\sqrt{x})^2 = x \quad \text{(for } x \geq 0\text{)} \]
*Note: The function is defined for \( x \geq 0 \) (including \( x = 0 \)).*

**Function h(x):**
\[ h(x) = x \quad \text{(for all } x \text{)} \]
*Note: This function is defined for all real numbers.*

### Step 2: Determine the Domains

- **Domain of \( f(x) \)**: \( x \in \mathbb{R} \setminus \{0\} \) (all real numbers except 0)
- **Domain of \( g(x) \)**: \( x \geq 0 \) (all non-negative real numbers)
- **Domain of \( h(x) \)**: \( x \in \mathbb{R} \) (all real numbers)

### Step 3: Compare the Functions

Now we compare the simplified forms of the functions along with their domains:

- **For \( f(x) \)** and **\( g(x) \)**:
- \( f(x) = x \) (for \( x \neq 0 \))
- \( g(x) = x \) (for \( x \geq 0 \))

They are not identical because \( f(x) \) is undefined at \( x = 0 \), while \( g(x) \) is defined at \( x = 0 \).

- **For \( f(x) \)** and **\( h(x) \)**:
- \( f(x) = x \) (for \( x \neq 0 \))
- \( h(x) = x \) (for all \( x \))

They are not identical because \( f(x) \) is undefined at \( x = 0 \).

- **For \( g(x) \)** and **\( h(x) \)**:
- \( g(x) = x \) (for \( x \geq 0 \))
- \( h(x) = x \) (for all \( x \))

They are not identical because \( g(x) \) is only defined for \( x \geq 0 \), while \( h(x) \) is defined for all \( x \).

### Conclusion

None of the functions \( f(x) \), \( g(x) \), and \( h(x) \) are identical because their domains differ.

### Final Answer
None of the functions are identical.

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