Find whether the following function are one-one or many -one & into or onto if `f:DtoR` where `D` is its domain
`f(x)=|x^(2)+5x+6|`

Explanation

To determine whether the function \( f(x) = |x^2 + 5x + 6| \) is one-one or many-one, and whether it is into or onto, we can follow these steps:

### Step 1: Analyze the function
The function given is \( f(x) = |x^2 + 5x + 6| \). First, we need to analyze the quadratic expression inside the absolute value.

### Step 2: Find the roots of the quadratic
To find the roots of the quadratic \( x^2 + 5x + 6 = 0 \), we can factor it:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) = 0 \]
The roots are \( x = -2 \) and \( x = -3 \).

### Step 3: Determine the behavior of the quadratic
Since the coefficient of \( x^2 \) is positive, the parabola opens upwards. The vertex of the parabola lies between the roots \( -3 \) and \( -2 \).

### Step 4: Sketch the graph of the quadratic
The graph of \( y = x^2 + 5x + 6 \) will be below the x-axis between the roots and above the x-axis outside this interval. The graph will touch the x-axis at \( x = -3 \) and \( x = -2 \).

### Step 5: Sketch the graph of the absolute value function
The graph of \( f(x) = |x^2 + 5x + 6| \) will reflect the part of the quadratic that is below the x-axis upwards. Thus, the graph will be:
- Above the x-axis for \( x < -3 \) and \( x > -2 \).
- Touching the x-axis at \( x = -3 \) and \( x = -2 \).

### Step 6: Determine if the function is one-one or many-one
To check if the function is one-one or many-one, we can use the horizontal line test:
- If a horizontal line intersects the graph at more than one point, the function is many-one.
- In this case, the graph of \( f(x) \) is U-shaped and intersects horizontal lines at multiple points (for example, at \( y = 1 \), it intersects at two different x-values).

Thus, the function \( f(x) \) is **many-one**.

### Step 7: Determine if the function is onto or into
The range of the function \( f(x) \) is from \( 0 \) to \( \infty \) (since the minimum value is \( 0 \) at the vertex and it goes to \( \infty \)). The codomain is \( \mathbb{R} \) (which is from \( -\infty \) to \( \infty \)).

Since the range \( [0, \infty) \) does not cover all real numbers, the function is **into**.

### Final Conclusion
- The function \( f(x) = |x^2 + 5x + 6| \) is **many-one** and **into**.