https://file.aiquickdraw.com/mathbot/file/1727412798844bq6b1qlr.jpg

Let's solve both parts of the given relations one by one.

### a) Relation: \( f(x) = -|x - 2| \)

Given that \( x = -2 \), we substitute this value into the function:

\[
f(-2) = -| -2 - 2 |
\]
\[
f(-2) = -| -4 | = -4
\]

So, for \( x = -2 \), the output is:
\[
f(x) = -4
\]

### b) Relation: \( f(x) = \frac{x - 1}{x} \)

Given that \( f(x) = 1 \), we set the equation equal to 1 and solve for \( x \):

\[
\frac{x - 1}{x} = 1
\]

Multiplying both sides by \( x \) (assuming \( x \neq 0 \)):

\[
x - 1 = x
\]

Subtracting \( x \) from both sides:

\[
-1 = 0
\]

This results in a contradiction, which means no real value of \( x \) can make \( f(x) = 1 \). Thus, this equation has **no solution** for \( x \).

### Summary:
- For part (a), \( f(-2) = -4 \).
- For part (b), there is no \( x \) such that \( f(x) = 1 \).

Let me know if you need further clarification or a detailed explanation!

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Here are 5 related questions you could consider:
1. What would the output be for other values of \( x \) in part (a)?
2. How does the absolute value affect the function in part (a)?
3. What is the behavior of the function \( f(x) = \frac{x-1}{x} \) as \( x \) approaches zero?
4. How would the result change in part (b) if \( f(x) = 0 \)?
5. Can you graph both functions to better visualize the relationships?

**Tip:** When dealing with absolute values, always break the problem into cases where the expression inside the absolute value is positive or negative.

For each of the following relations, determine the missing inputs or outputs.

Solve for missing inputs or outputs in relations involving absolute value and rational functions. Understand how to substitute values into functions and solve equations. Suitable for students in Grades 9-12.

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