If `f(x)=sqrt(x^2+a x+4)` is defined for all `x ,` then find the values of `adot`

Explanation

To solve the problem, we need to ensure that the function \( f(x) = \sqrt{x^2 + ax + 4} \) is defined for all \( x \). This means that the expression inside the square root must be non-negative for all values of \( x \).

### Step-by-Step Solution:

1. **Set up the inequality**:
We need to ensure that:
\[ x^2 + ax + 4 \geq 0 \quad \text{for all } x \]

2. **Identify the discriminant**:
For a quadratic expression \( Ax^2 + Bx + C \) to be non-negative for all \( x \), its discriminant must be less than or equal to zero. The discriminant \( D \) for the quadratic \( x^2 + ax + 4 \) is given by:
\[ D = b^2 - 4ac = a^2 - 4(1)(4) = a^2 - 16 \]

3. **Set the discriminant condition**:
We require:
\[ a^2 - 16 \leq 0 \]

4. **Solve the inequality**:
Rearranging the inequality gives:
\[ a^2 \leq 16 \]
Taking the square root of both sides, we find:
\[ -4 \leq a \leq 4 \]

5. **Conclusion**:
Therefore, the values of \( a \) for which the function \( f(x) \) is defined for all \( x \) are:
\[ a \in [-4, 4] \]