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Let's solve the given problems step by step.

The functions provided are:
\[ f(x) = |x| \]
\[ g(x) = x - 1 \]

### Problem 1:
Given \( x \) is a random real number from the interval \([-3, 4]\) and \( y = 2 \), find the probability that \( f(x) \geq |g(y)| + 1 \).

1. Compute \( |g(y)| \):
\[ g(y) = g(2) = 2 - 1 = 1 \]
\[ |g(y)| = |1| = 1 \]

2. The inequality becomes:
\[ f(x) \geq 1 + 1 \]
\[ f(x) \geq 2 \]

3. Since \( f(x) = |x| \), we have:
\[ |x| \geq 2 \]

4. This implies:
\[ x \leq -2 \quad \text{or} \quad x \geq 2 \]

5. The interval \([-3, 4]\) can be divided as follows:
\[ [-3, -2] \cup [-2, 2] \cup [2, 4] \]

6. The length of the intervals where \( |x| \geq 2 \):
\[ [-3, -2] \quad \text{and} \quad [2, 4] \]

7. The total length of these intervals:
\[ (-2 - (-3)) + (4 - 2) = 1 + 2 = 3 \]

8. The total length of the interval \([-3, 4]\) is:
\[ 4 - (-3) = 7 \]

9. The probability is:
\[ \frac{3}{7} \]

So, the probability for part (1) is \(\frac{3}{7}\).

### Problem 2:
Given \( x \) and \( y \) are random real numbers from the interval \([0, 4]\), find the probability that \( f^2(x) + g^2(y) \leq 1 \).

1. Rewrite the inequality using the given functions:
\[ f(x) = |x| \quad \text{and} \quad g(y) = y - 1 \]
\[ f^2(x) + g^2(y) = x^2 + (y - 1)^2 \]

2. We need to find the region where:
\[ x^2 + (y - 1)^2 \leq 1 \]

3. This represents a circle with center \((0, 1)\) and radius 1.

4. The region of interest is the intersection of this circle with the square \([0, 4] \times [0, 4]\).

5. The circle \( x^2 + (y - 1)^2 = 1 \) intersects the y-axis at \( y = 0 \) and \( y = 2 \), meaning the circle is entirely within the interval \( 0 \leq y \leq 2 \).

6. Calculate the area of the circle segment within the square:
   - The circle is entirely within the square for \( 0 \leq x \leq 1 \).
   - For \( x = 1 \), the maximum value of \( y \) is at the top of the circle \( y = 2 \).

7. The area within the square and circle is a quarter of the circle:
\[ \frac{\pi

Explore advanced probability problems involving absolute value functions and inequalities. Solve for probabilities where the absolute value function and linear functions intersect within given intervals. Suitable for advanced high school students.

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