The function `f:(-oo, 1] rarr (0, e^(5)]` defined as `f(x)=e^(x^(3)+2)` is

Options

A. Many one and onto
B. Many one and into
C. one - one and onto
D. one - one and into

Answer

To determine the properties of the function \( f(x) = e^{x^3 + 2} \) defined from the domain \( (-\infty, 1] \) to the codomain \( (0, e^5] \), we need to analyze whether it is one-one (injective) and onto (surjective).

### Step 1: Check if the function is one-one (injective)

A function is one-one if different inputs produce different outputs. To check this, we can analyze the derivative of the function.

1. **Find the derivative of \( f(x) \)**:
\[ f'(x) = \frac{d}{dx}(e^{x^3 + 2}) = e^{x^3 + 2} \cdot \frac{d}{dx}(x^3 + 2) = e^{x^3 + 2} \cdot 3x^2 \]

2. **Analyze the sign of \( f'(x) \)**:
- Since \( e^{x^3 + 2} > 0 \) for all \( x \) and \( 3x^2 \geq 0 \) (with equality only at \( x = 0 \)), \( f'(x) \) is non-negative.
- \( f'(x) = 0 \) only when \( x = 0 \).

3. **Determine the behavior of \( f(x) \)**:
- For \( x < 0 \), \( f'(x) > 0 \) (the function is increasing).
- At \( x = 0 \), \( f'(0) = 0 \) (local minimum).
- For \( x > 0 \) (up to 1), \( f'(x) > 0 \) (the function continues increasing).

Since the function is increasing for all \( x \) in the domain, it is one-one.

### Step 2: Check if the function is onto (surjective)

A function is onto if every element in the codomain has a pre-image in the domain.

1. **Determine the range of \( f(x) \)**:
- As \( x \to -\infty \), \( x^3 \to -\infty \), thus \( f(x) \to e^{-\infty} = 0 \).
- At \( x = 1 \), \( f(1) = e^{1^3 + 2} = e^3 \).

2. **Analyze the range**:
- The function \( f(x) \) is continuous and strictly increasing from \( 0 \) to \( e^3 \) as \( x \) goes from \( -\infty \) to \( 1 \).
- Therefore, the range of \( f(x) \) is \( (0, e^3] \).

3. **Compare the range with the codomain**:
- The codomain is \( (0, e^5] \), which is larger than the range \( (0, e^3] \).
- Since not every element in \( (0, e^5] \) can be achieved (for example, values between \( e^3 \) and \( e^5 \)), the function is not onto.

### Conclusion

The function \( f(x) = e^{x^3 + 2} \) is one-one but not onto.