Which one of the following functions is not one-one?

Options

A. `f:(-1,oo) to R " given by "f(x)=x^(2)+2x`
B. `g:(1,oo) to R " given by "g(x)=e^(x^(3)-3x+2)`
C. `h: R to R "given by h(x)=2^(x^(x-1))`
D. `phi,(-oo,0)to R "given by" phi(x)=(x^(2))/(x^(2)+1)`

Explanation

To determine which function is not one-one among the given options, we will analyze each function step by step.

### Step 1: Analyze Option A
**Function:** \( f(x) = x^2 + 2x \)
**Domain:** \( x \in [-1, \infty) \)

1. **Differentiate the function:**
\[ f'(x) = 2x + 2 \]

2. **Set the derivative to zero to find critical points:**
\[ 2x + 2 = 0 \implies x = -1 \]

3. **Evaluate the sign of the derivative:**
- For \( x < -1 \), \( f'(x) < 0 \) (decreasing)
- For \( x > -1 \), \( f'(x) > 0 \) (increasing)

Since the function decreases until \( x = -1 \) and then increases, it is not one-one over the entire interval \( [-1, \infty) \).

### Step 2: Analyze Option B
**Function:** \( g(x) = e^{x^3 - 3x + 2} \)
**Domain:** \( x \in [1, \infty) \)

1. **Differentiate the function:**
\[ g'(x) = e^{x^3 - 3x + 2} \cdot (3x^2 - 3) \]

2. **Set the derivative to zero:**
\[ 3x^2 - 3 = 0 \implies x^2 = 1 \implies x = 1 \text{ (only valid in the domain)} \]

3. **Evaluate the sign of the derivative:**
- For \( x > 1 \), \( g'(x) > 0 \) (increasing)

Since \( g(x) \) is increasing for \( x \geq 1 \), it is one-one.

### Step 3: Analyze Option C
**Function:** \( h(x) \) (not explicitly given, but assumed from context)
**Domain:** \( x \in \mathbb{R} \)

1. **Differentiate the function:**
\[ h'(x) > 0 \text{ for all } x > \frac{1}{2} \]
\[ h'(x) < 0 \text{ for all } x < \frac{1}{2} \]

Since \( h'(x) \) changes sign, it indicates that \( h(x) \) is not one-one.

### Step 4: Analyze Option D
**Function:** \( \phi(x) = \frac{x^2}{x^2 + 1} \)
**Domain:** \( x \in (-\infty, 0) \)

1. **Differentiate the function:**
\[ \phi'(x) = \frac{(x^2 + 1)(2x) - (x^2)(2x)}{(x^2 + 1)^2} = \frac{2x}{(x^2 + 1)^2} \]

2. **Evaluate the sign of the derivative:**
- For \( x < 0 \), \( \phi'(x) < 0 \) (decreasing)

Since \( \phi(x) \) is decreasing for \( x < 0 \), it is one-one.

### Conclusion
From the analysis:
- **Option A** is not one-one.
- **Option B** is one-one.
- **Option C** is not one-one.
- **Option D** is one-one.

Thus, the functions that are not one-one are **Option A** and **Option C**. However, since the question asks for one function that is not one-one, we can conclude that **Option C** is the correct answer.

### Final Answer
**Option C** is not one-one.