Consider the following functions:
1. `f(x)=x^(3),x""inRR`
2. `f(x)=sinx,0ltxlt2pi`
3. `f(x)=e^(x ),x""inRR`
Which of the above function have inverse defined on their ranges?

Options

A. 1 and 2 olny
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Answer

To determine which of the given functions have inverses defined on their ranges, we will analyze each function step by step.

### Step 1: Analyze the function \( f(x) = x^3 \) for \( x \in \mathbb{R} \)

1. **Domain**: The domain of \( f(x) = x^3 \) is all real numbers \( \mathbb{R} \).
2. **Range**: The range of \( f(x) = x^3 \) is also all real numbers \( \mathbb{R} \) because as \( x \) takes any real value, \( x^3 \) will cover all real values.
3. **Invertibility**: Since \( f(x) = x^3 \) is a one-to-one function (it is strictly increasing), it has an inverse. The inverse function is \( f^{-1}(y) = y^{1/3} \).

### Conclusion for Step 1:
- The function \( f(x) = x^3 \) has an inverse defined on its range.

### Step 2: Analyze the function \( f(x) = \sin x \) for \( 0 < x < 2\pi \)

1. **Domain**: The domain of \( f(x) = \sin x \) is \( (0, 2\pi) \).
2. **Range**: The range of \( f(x) = \sin x \) over this interval is \( [-1, 1] \).
3. **Invertibility**: The sine function is not one-to-one over the interval \( (0, 2\pi) \) because it repeats values (e.g., \( \sin(\pi/2) = 1 \) and \( \sin(3\pi/2) = -1 \)). Therefore, it does not have an inverse that is defined on the range \( [-1, 1] \).

### Conclusion for Step 2:
- The function \( f(x) = \sin x \) does not have an inverse defined on its range.

### Step 3: Analyze the function \( f(x) = e^x \) for \( x \in \mathbb{R} \)

1. **Domain**: The domain of \( f(x) = e^x \) is all real numbers \( \mathbb{R} \).
2. **Range**: The range of \( f(x) = e^x \) is \( (0, \infty) \) because \( e^x \) is always positive for all real \( x \).
3. **Invertibility**: The function \( f(x) = e^x \) is one-to-one (it is strictly increasing), and it has an inverse. The inverse function is \( f^{-1}(y) = \ln y \).

### Conclusion for Step 3:
- The function \( f(x) = e^x \) has an inverse defined on its range.

### Final Conclusion:
- The functions that have inverses defined on their ranges are:
1. \( f(x) = x^3 \)
2. \( f(x) = e^x \)

Thus, the answer is that the functions \( f(x) = x^3 \) and \( f(x) = e^x \) have inverses defined on their ranges.