The function `f(x)=x^(2)+bx+c`, where b and c are real constants, describes

Options

A. one-one mapping
B. onto mapping
C. not one-one but onto mapping
D. Neither one-one nor onto mapping

Explanation

To solve the problem regarding the function \( f(x) = x^2 + bx + c \), we need to analyze its properties in terms of mapping (one-one and onto). Here’s a step-by-step breakdown:

### Step 1: Identify the Type of Function
The function given is a quadratic function of the form \( f(x) = ax^2 + bx + c \) where \( a = 1 \), \( b \) and \( c \) are real constants. Since \( a > 0 \), this function represents a parabola that opens upwards.

**Hint:** Remember that the general form of a quadratic function is \( ax^2 + bx + c \).

### Step 2: Determine the Vertex
The vertex of the parabola can be found using the formula for the x-coordinate of the vertex, which is given by:
\[ x_v = -\frac{b}{2a} \]
Substituting \( a = 1 \):
\[ x_v = -\frac{b}{2} \]

**Hint:** The vertex of a parabola gives the minimum or maximum point depending on the direction it opens.

### Step 3: Calculate the Minimum Value
The minimum value of the function occurs at the vertex. To find the minimum value, substitute \( x_v \) back into the function:
\[ f\left(-\frac{b}{2}\right) = \left(-\frac{b}{2}\right)^2 + b\left(-\frac{b}{2}\right) + c \]
This simplifies to:
\[ f\left(-\frac{b}{2}\right) = \frac{b^2}{4} - \frac{b^2}{2} + c = -\frac{b^2}{4} + c \]

**Hint:** The minimum value of a quadratic function can help determine its range.

### Step 4: Analyze the Range
Since the parabola opens upwards, the range of the function starts from the minimum value calculated above and goes to infinity:
\[ \text{Range} = \left[-\frac{b^2}{4} + c, \infty\right) \]

**Hint:** The range of a function tells us the possible output values.

### Step 5: Check for One-One Mapping
A function is one-one (injective) if it never takes the same value twice for different inputs. Since the parabola opens upwards and has a minimum point, it will have the same output for two different inputs (one on each side of the vertex). Therefore, the function is not one-one.

**Hint:** Graphing the function can help visualize whether it is one-one.

### Step 6: Check for Onto Mapping
A function is onto (surjective) if every possible output in the codomain is achieved by some input from the domain. Since the range of \( f(x) \) is \( \left[-\frac{b^2}{4} + c, \infty\right) \), it does not cover all real numbers (specifically, it does not reach values below \( -\frac{b^2}{4} + c \)). Thus, the function is not onto.

**Hint:** Consider the codomain of the function when determining if it is onto.

### Conclusion
The function \( f(x) = x^2 + bx + c \) is neither one-one nor onto.

**Final Answer:** The function is neither one-one nor onto.