Let's analyze the expression [tex]\( P = \frac{4 - \sqrt{8x - 4x^2}}{4 + x} \)[/tex] to determine the conditions under which it is real and under which it is undefined.
### 1.3.1 When is [tex]\(P\)[/tex] real?
For [tex]\( P \)[/tex] to be real, the expression inside the square root, [tex]\( 8x - 4x^2 \)[/tex], must be non-negative. This means we need to solve the inequality:
[tex]\[ 8x - 4x^2 \geq 0 \][/tex]
Let's factor the quadratic expression:
[tex]\[ 8x - 4x^2 = 4x(2 - x) \][/tex]
Thus, the inequality becomes:
[tex]\[ 4x(2 - x) \geq 0 \][/tex]
We solve this by considering the sign of the product [tex]\( 4x(2 - x) \)[/tex]:
1. The factor [tex]\( 4x \)[/tex] is non-negative when [tex]\( x \geq 0 \)[/tex].
2. The factor [tex]\( 2 - x \)[/tex] is non-negative when [tex]\( x \leq 2 \)[/tex].
Combining these two conditions, we find that:
[tex]\[ 0 \leq x \leq 2 \][/tex]
Therefore, the expression [tex]\( P \)[/tex] is real for:
[tex]\[ 0 \leq x \leq 2 \][/tex]
### 1.3.2 When is [tex]\(P\)[/tex] undefined?
The expression [tex]\( P \)[/tex] will be undefined when the denominator is zero. Hence, we need to solve the equation:
[tex]\[ 4 + x = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -4 \][/tex]
Therefore, the expression [tex]\( P \)[/tex] is undefined for:
[tex]\[ x = -4 \][/tex]
### Summary
1. [tex]\( P \)[/tex] is real for: [tex]\( 0 \leq x \leq 2 \)[/tex]
2. [tex]\( P \)[/tex] is undefined for: [tex]\( x = -4 \)[/tex]
