Find the real values of x for which the function `f(x) = cos^(-1) sqrt(x^(2) + 3 x + 1) + cos^(-1) sqrt(x^(2) + 3x)` is defined

Explanation

To find the real values of \( x \) for which the function

\[ f(x) = \cos^{-1} \sqrt{x^2 + 3x + 1} + \cos^{-1} \sqrt{x^2 + 3x} \]

is defined, we need to ensure that the expressions inside the inverse cosine functions are valid. Specifically, we need to check the conditions under which \( \sqrt{x^2 + 3x + 1} \) and \( \sqrt{x^2 + 3x} \) are both defined and lie within the interval \([0, 1]\).

### Step 1: Determine the conditions for \( \sqrt{x^2 + 3x + 1} \)

The expression \( \sqrt{x^2 + 3x + 1} \) is defined when:

\[ x^2 + 3x + 1 \geq 0 \]

To find the roots of the quadratic equation \( x^2 + 3x + 1 = 0 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 1, b = 3, c = 1 \).

Calculating the discriminant:

\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]

The roots are:

\[ x = \frac{-3 \pm \sqrt{5}}{2} \]

### Step 2: Determine the intervals for \( x^2 + 3x + 1 \geq 0 \)

The roots divide the number line into intervals. We need to test the sign of \( x^2 + 3x + 1 \) in these intervals:

1. \( (-\infty, \frac{-3 - \sqrt{5}}{2}) \)
2. \( \left(\frac{-3 - \sqrt{5}}{2}, \frac{-3 + \sqrt{5}}{2}\right) \)
3. \( \left(\frac{-3 + \sqrt{5}}{2}, \infty\right) \)

By testing a point from each interval, we can determine where the quadratic is non-negative.

### Step 3: Determine the conditions for \( \sqrt{x^2 + 3x} \)

Similarly, for \( \sqrt{x^2 + 3x} \), we require:

\[ x^2 + 3x \geq 0 \]

Factoring gives:

\[ x(x + 3) \geq 0 \]

The roots are \( x = 0 \) and \( x = -3 \). The intervals to test are:

1. \( (-\infty, -3) \)
2. \( [-3, 0] \)
3. \( (0, \infty) \)

### Step 4: Combine the conditions

We need to find the intersection of the intervals where both conditions hold true.

1. From \( x^2 + 3x + 1 \geq 0 \), we find the intervals where it is non-negative.
2. From \( x^2 + 3x \geq 0 \), we find the intervals where it is non-negative.

### Final Step: Identify the real values of \( x \)

After analyzing both conditions, we conclude that the function \( f(x) \) is defined for:

\[ x \in (-\infty, -3] \cup [0, \infty) \]