Which of the following is (are) incorrect ?

Answer

To determine which of the statements is incorrect, we will analyze each option step by step.

### Step 1: Analyze Option A
**Statement:** If \( f(x) = \sin x \) and \( g(x) = \sin x \), then the range of \( g(f(x)) \) is \([-1, 1]\).

1. **Finding the range of \( f(x) \):**
- Since \( f(x) = \sin x \), the range of \( f(x) \) is \([-1, 1]\).

2. **Finding \( g(f(x)) \):**
- Since \( g(x) = \sin x \), we have \( g(f(x)) = g(\sin x) = \sin(\sin x) \).

3. **Finding the range of \( g(f(x)) \):**
- The output of \( \sin x \) lies between \([-1, 1]\).
- Thus, \( g(f(x)) = \sin(\sin x) \) will take values in the range of \( \sin x \), which is between \(\sin(-1)\) and \(\sin(1)\).
- The values of \(\sin(-1)\) and \(\sin(1)\) are approximately \(-0.8415\) and \(0.8415\) respectively.

4. **Conclusion for Option A:**
- Therefore, the range of \( g(f(x)) \) is not \([-1, 1]\), making this statement incorrect.

### Step 2: Analyze Option B
**Statement:** If \( f(x) = \frac{2011 - 2012}{2011 - x^{2012}} \), then \( f(f(2)) = \frac{1}{2} \).

1. **Calculate \( f(2) \):**
- Substitute \( x = 2 \):
\[ f(2) = \frac{2011 - 2012}{2011 - 2^{2012}} = \frac{-1}{2011 - 2^{2012}} \]

2. **Calculate \( f(f(2)) \):**
- Let \( y = f(2) = \frac{-1}{2011 - 2^{2012}} \).
- Now substitute \( x = y \) into \( f(x) \):
\[ f(y) = \frac{2011 - 2012}{2011 - \left(\frac{-1}{2011 - 2^{2012}}\right)^{2012}} \]
- After simplification, it can be shown that \( f(f(2)) \) does not equal \( \frac{1}{2} \).

3. **Conclusion for Option B:**
- Thus, this statement is also incorrect.

### Step 3: Analyze Option C
**Statement:** The function defined as \( f(x) = x^2 + 4x + 30 \) is not surjective.

1. **Finding the range of \( f(x) \):**
- Rewrite \( f(x) \) in vertex form:
\[ f(x) = (x + 2)^2 + 26 \]
- The minimum value occurs at \( x = -2 \), which gives \( f(-2) = 26 \).

2. **Conclusion for Option C:**
- Since the minimum value of \( f(x) \) is 26, the range of \( f(x) \) is \([26, \infty)\), which does not cover all real numbers. Therefore, \( f(x) \) is not surjective.

### Final Conclusion
- The incorrect options are A and B, while C is correct.

### Summary of Incorrect Options
- **Incorrect Options:** A and B