https://file.aiquickdraw.com/mathbot/file/172678780844936kmmujo.jpg

Let's go through the functions you provided in the image and evaluate them. Here is the breakdown of each one:

1. **\( f(x) = 2x - 11, \, x = 10 \)**
   \[
   f(10) = 2(10) - 11 = 20 - 11 = 9
   \]
   **Answer**: \( f(10) = 9 \)

2. **\( f(x) = 6x^2 + 4x - 7, \, x = 2 \)**
   \[
   f(2) = 6(2^2) + 4(2) - 7 = 6(4) + 8 - 7 = 24 + 8 - 7 = 25
   \]
   **Answer**: \( f(2) = 25 \)

3. **\( F(x) = 3x^2 + 5x + 1, \, x = 2 \)**
   \[
   F(2) = 3(2^2) + 5(2) + 1 = 3(4) + 10 + 1 = 12 + 10 + 1 = 23
   \]
   **Answer**: \( F(2) = 23 \)

4. **\( f(k) = 5k + 1, \, k = (k+2) \)**
   This one is a bit different, as it's a function where \( k \) is substituted with \( k+2 \):
   \[
   f(k) = 5(k+2) + 1 = 5k + 10 + 1 = 5k + 11
   \]
   **Answer**: \( f(k) = 5k + 11 \)

5. **\( f(x) = x - 2|2x - 7|, \, x = 7 \)**
   First, evaluate the absolute value:
   \[
   f(7) = 7 - 2|2(7) - 7| = 7 - 2|14 - 7| = 7 - 2|7| = 7 - 2(7) = 7 - 14 = -7
   \]
   **Answer**: \( f(7) = -7 \)

Would you like any further explanation or details?

Here are some follow-up questions you can explore:

1. How do you handle functions that include absolute values in different cases (e.g., when the expression inside is negative)?
2. What happens to the domain of a function if you introduce a square root or a fraction?
3. How would you solve for unknown values inside a function (e.g., find \( k \) in \( f(k) = 5k + 1 \))?
4. Can you describe how polynomials like \( 3x^2 + 5x + 1 \) behave as \( x \) grows larger?
5. What are some real-life applications where absolute values in functions matter?

**Tip**: Always start by simplifying the inner expressions (like absolute values or powers) before substituting specific values.

Evaluate the following functions: f(x) = 2x - 11 at x = 10, f(x) = 6x^2 + 4x - 7 at x = 2, F(x) = 3x^2 + 5x + 1 at x = 2, f(k) = 5k + 1 at k = (k+2), f(x) = x - 2|2x - 7| at x = 7.

Learn how to evaluate functions including linear, quadratic, and absolute value functions with a detailed step-by-step solution. This problem involves the evaluation of f(x) = 2x - 11 at x = 10, f(x) = 6x^2 + 4x - 7 at x = 2, F(x) = 3x^2 + 5x + 1 at x = 2, and more. Suitable for students in Grades 9-12.

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