https://file.aiquickdraw.com/mathbot/file/1726232793063hv0k7agp.jpg

Let's analyze each problem in the image one by one:

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**a. \( f(x) = |x| \), find \( f(8) \)**

- The function is the absolute value function. Therefore:
  \[
  f(8) = |8| = 8
  \]

---

**b. \( f(x) = 3 - \sqrt{x} \), find \( f(x) = 2 \)**

- Set \( f(x) = 2 \):
  \[
  3 - \sqrt{x} = 2
  \]
  Subtract 3 from both sides:
  \[
  -\sqrt{x} = -1
  \]
  Multiply both sides by -1:
  \[
  \sqrt{x} = 1
  \]
  Square both sides:
  \[
  x = 1
  \]

So, \( x = 1 \) is the solution.

---

**c. \( f(k) = \frac{k}{2} + 1 \), find \( f(-6) \)**

- Substitute \( k = -6 \) into the function:
  \[
  f(-6) = \frac{-6}{2} + 1 = -3 + 1 = -2
  \]

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**d. \( f(x) = \sqrt{x - 5} \), find \( f(3) \)**

- Substitute \( x = 3 \) into the function:
  \[
  f(3) = \sqrt{3 - 5} = \sqrt{-2}
  \]
  Since the square root of a negative number is not real, there is **no real solution**.

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### Summary of Results:
- a) \( f(8) = 8 \)
- b) \( x = 1 \)
- c) \( f(-6) = -2 \)
- d) No real solution for \( f(3) \)

Would you like more details on any of the steps, or have any questions?

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Here are 5 related questions to further explore this topic:
1. What is the domain of the function \( f(x) = \sqrt{x - 5} \)?
2. How would the result change for part b if \( f(x) = 4 - \sqrt{x} \) instead?
3. What is the graph of \( f(x) = 3 - \sqrt{x} \) and what does it indicate about its range?
4. How can you rewrite \( f(k) = \frac{k}{2} + 1 \) as an inverse function?
5. What are the properties of absolute value functions in terms of continuity and differentiability?

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**Tip:** For functions involving square roots, always check the domain to ensure the expression under the root is non-negative!

Find the corresponding inputs or outputs for the following functions. If there is no solution, explain why not. Be careful: In some cases, there may be no solution or more than one possible solution.

Explore solving various function types, including absolute value, square root, and linear functions. This problem covers algebraic concepts such as function domains, square roots, and absolute value properties. Suitable for Grades 8-10.

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