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 will 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 modulus.

### Step 2: Factor the quadratic
We can factor \( x^2 + 5x + 6 \):
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
This means the quadratic has roots at \( x = -2 \) and \( x = -3 \).

### Step 3: Determine the behavior of the function
Since the quadratic opens upwards (the coefficient of \( x^2 \) is positive), it will take negative values between its roots and positive values outside of that interval. Therefore, the function \( f(x) \) will be:
- \( f(x) = 0 \) at \( x = -2 \) and \( x = -3 \)
- Positive for \( x < -3 \) and \( x > -2 \)

### Step 4: Check for one-one or many-one
To check if the function is one-one, we need to see if different inputs can produce the same output. From our analysis:
- \( f(-3) = 0 \)
- \( f(-2) = 0 \)

Since both \( -2 \) and \( -3 \) give the same output (0), the function is **not one-one**. Therefore, it is **many-one**.

### Step 5: Determine if the function is onto or into
The range of \( f(x) \) is from \( 0 \) to \( \infty \) (since the modulus ensures non-negative outputs). The co-domain is the set of all real numbers \( \mathbb{R} \).

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

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

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