If f(x)=x, g(x)=`sqrt(x^2)` and h(x)=`x^2/x`, then the set of values for which the given functions f(x), g(x) and h(x) are identical is :

Options

A. R-{0}
B. R
C. `R^+`
D. `R^+ cup {0}`

Answer

To solve the problem, we need to analyze the functions \( f(x) = x \), \( g(x) = \sqrt{x^2} \), and \( h(x) = \frac{x^2}{x} \) to determine the set of values for which these functions are identical.

### Step-by-Step Solution:

1. **Identify the functions:**
- \( f(x) = x \)
- \( g(x) = \sqrt{x^2} \)
- \( h(x) = \frac{x^2}{x} \)

2. **Determine the domain of each function:**
- **For \( f(x) = x \)**:
- The domain is all real numbers, \( x \in \mathbb{R} \).

- **For \( g(x) = \sqrt{x^2} \)**:
- The expression inside the square root must be non-negative:
\[ x^2 \geq 0 \]
- This is true for all \( x \in \mathbb{R} \). Therefore, the domain is also \( x \in \mathbb{R} \).

- **For \( h(x) = \frac{x^2}{x} \)**:
- This function is defined for \( x \neq 0 \) (since division by zero is undefined).
- Therefore, the domain is \( x \in \mathbb{R} \setminus \{0\} \).

3. **Find the set of values for which the functions are identical:**
- **For \( f(x) \) and \( g(x) \)**:
- \( g(x) = \sqrt{x^2} = |x| \). Thus, \( f(x) = x \) and \( g(x) = |x| \) are equal when \( x \geq 0 \) (i.e., \( f(x) = g(x) \) for \( x \geq 0 \)).
- For \( x < 0 \), \( f(x) \) is negative while \( g(x) \) is positive, so they are not equal.

- **For \( h(x) \)**:
- \( h(x) = \frac{x^2}{x} = |x| \) for \( x \neq 0 \). Similar to \( g(x) \), \( h(x) \) is equal to \( x \) when \( x \geq 0 \).

4. **Combine the conditions:**
- The functions \( f(x) \), \( g(x) \), and \( h(x) \) are identical when \( x \geq 0 \) and \( x \neq 0 \). Thus, the set of values for which all three functions are identical is:
\[ x > 0 \]

5. **Conclusion:**
- The set of values for which the functions \( f(x) \), \( g(x) \), and \( h(x) \) are identical is \( (0, \infty) \).