Which pairs of function is identical ?

Options

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

Answer

To determine which pairs of functions are identical, we need to analyze the domain and range of each function given in the options. Identical functions must have the same domain and range.

### Step-by-Step Solution:

1. **Analyze Option 1:**
- Function \( f(x) = \sqrt{x^2} \)
- The expression \( \sqrt{x^2} \) simplifies to \( |x| \).
- The domain of \( f(x) \) is \( x \in [0, \infty) \) because the square root function is defined for non-negative values.
- Function \( g(x) = x \)
- The domain of \( g(x) \) is \( x \in (-\infty, \infty) \).
- **Conclusion for Option 1:** The domains are different; hence, they are not identical.

2. **Analyze Option 2:**
- Function \( f(x) = \sin^2(x) + \cos^2(x) \)
- This is a well-known trigonometric identity that equals 1 for all \( x \).
- Therefore, the domain of \( f(x) \) is \( x \in (-\infty, \infty) \) and the range is \( \{1\} \).
- Function \( g(x) = 1 \)
- The domain of \( g(x) \) is also \( x \in (-\infty, \infty) \) and the range is \( \{1\} \).
- **Conclusion for Option 2:** Both functions have the same domain and range, so they are identical.

3. **Analyze Option 3:**
- Function \( f(x) = \frac{x}{x} \)
- This function is defined for all \( x \) except \( x = 0 \) (since division by zero is undefined).
- Therefore, the domain of \( f(x) \) is \( x \in (-\infty, 0) \cup (0, \infty) \).
- Function \( g(x) = 1 \)
- The domain of \( g(x) \) is \( x \in (-\infty, \infty) \).
- **Conclusion for Option 3:** The domains are different; hence, they are not identical.

### Final Conclusion:
The only pair of functions that are identical is from **Option 2**: \( f(x) = \sin^2(x) + \cos^2(x) \) and \( g(x) = 1 \).