We are given a piecewise function $f(x)$ and asked to determine the values of $a$ and $b$ based on the following conditions:

1. When $x = 4$, $f(x) = a + b$.
2. When $x < 4$, $f(x) = \frac{x - 4}{|x - 4|} + a$.
3. When $x > 4$, $f(x) = \frac{x - 4}{|x - 4|} + b$.

Step 1: Analyze the first condition

We are told that when $x = 4$, the function equals $a + b$. Therefore:

$f(4) = a + b.$

Step 2: Analyze the second condition ($x < 4$)

For $x < 4$, we have:

$f(x) = \frac{x - 4}{|x - 4|} + a.$

When $x < 4$, $x - 4$ is negative, so $|x - 4| = -(x - 4)$. Therefore:

$f(x) = \frac{x - 4}{-(x - 4)} + a = -1 + a.$

Thus, for $x < 4$, we have $f(x) = a - 1$.

Step 3: Analyze the third condition ($x > 4$)

For $x > 4$, we have:

$f(x) = \frac{x - 4}{|x - 4|} + b.$

When $x > 4$, $x - 4$ is positive, so $|x - 4| = x - 4$. Therefore:

$f(x) = \frac{x - 4}{x - 4} + b = 1 + b.$

Thus, for $x > 4$, we have $f(x) = b + 1$.

Step 4: Continuity and value matching

For the function to be continuous at $x = 4$, the values of $f(x)$ from both sides must be equal to $f(4)$.

• As $x$ approaches 4 from the left ($x < 4$), $f(x) = a - 1$. Thus:

$\lim_{x \to 4^-} f(x) = a - 1.$

• As $x$ approaches 4 from the right ($x > 4$), $f(x) = b + 1$. Thus:

$\lim_{x \to 4^+} f(x) = b + 1.$

For continuity at $x = 4$, the function values from both sides must be equal to $f(4) = a + b$.

Thus, we must have:

$a - 1 = a + b \quad \text{and} \quad b + 1 = a + b.$

Step 5: Solve the system of equations

Now, solve the system of equations:

1. $a - 1 = a + b$ simplifies to:

$-1 = b.$

2. $b + 1 = a + b$ simplifies to:

$1 = a.$

Step 6: Conclusion

From the above, we find:

$a = 1 \quad \text{and} \quad b = -1.$

Final Answer:

The values of $a$ and $b$ are:

$a = 1 \quad \text{and} \quad b = -1.$

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Would you like more details or explanations on any of these steps? Here are 5 related questions:

1. How would the function change if the conditions for $x < 4$ and $x > 4$ were swapped?
2. What happens if we remove the continuity condition at $x = 4$?
3. How would the value of $f(4)$ change if $a$ and $b$ were different?
4. Can piecewise functions always be solved for $a$ and $b$ in this way?
5. What other types of conditions can be applied to piecewise functions?

Tip: Always check for continuity in piecewise functions before solving.