Function is said to be onto if range is same as co - domain otherwise it is into. Function is said to be one - one if for all `x_(1) ne x_(2) rArr f(x_(1)) ne f(x_(2))` otherwise it is many one.
Function `f:R rarr R, f(x)=x^(2)+x`, is

Options

A. one - one onto
B. one - one into
C. many one onto
D. many one into

Answer

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^2 + x \) is one-one or many-one, and whether it is onto or into, we will follow these steps:

### Step 1: Analyze the function
The function \( f(x) = x^2 + x \) is a quadratic function. Quadratic functions typically have a parabolic shape.

**Hint:** Quadratic functions can be analyzed by completing the square or finding their vertex.

### Step 2: Complete the square
We can rewrite \( f(x) \) by completing the square:
\[ f(x) = x^2 + x = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} \]

**Hint:** Completing the square helps us find the vertex of the parabola, which gives us the minimum or maximum value.

### Step 3: Identify the vertex
The vertex of the parabola represented by \( f(x) \) is at \( x = -\frac{1}{2} \). The minimum value of \( f(x) \) occurs at this point:
\[ f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2} + \frac{1}{2}\right)^2 - \frac{1}{4} = -\frac{1}{4} \]

**Hint:** The vertex of a parabola gives the minimum or maximum value of the function.

### Step 4: Determine the range
Since the parabola opens upwards (the coefficient of \( x^2 \) is positive), the range of \( f(x) \) is:
\[ \text{Range} = \left[-\frac{1}{4}, \infty\right) \]

**Hint:** The range of a function is the set of all possible output values.

### Step 5: Determine the co-domain
The co-domain of the function is \( \mathbb{R} \), which is all real numbers:
\[ \text{Co-domain} = \mathbb{R} = (-\infty, \infty) \]

**Hint:** The co-domain is the set of values that the function could potentially output, regardless of whether it actually does.

### Step 6: Check if the function is onto
A function is onto if its range is equal to its co-domain. Here, the range \( \left[-\frac{1}{4}, \infty\right) \) is not equal to the co-domain \( \mathbb{R} \). Therefore, the function is not onto; it is into.

**Hint:** Compare the range and co-domain to determine if the function is onto.

### Step 7: Check if the function is one-one
To determine if the function is one-one, we can use the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, the function is not one-one. Since \( f(x) \) is a parabola that opens upwards, it will intersect any horizontal line above its minimum value at two points. Therefore, the function is many-one.

**Hint:** The horizontal line test is a visual method to determine if a function is one-one.

### Conclusion
The function \( f(x) = x^2 + x \) is many-one and into.

### Final Answer
- **Many-one:** Yes
- **Into:** Yes