https://file.aiquickdraw.com/mathbot/file/1727173827835c9mlsbgx.jpg

Let's go through the questions from the image and provide solutions.

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1. A function is ______ a relation, but not all relations are functions.
   - **Answer**: c. always

2. What is the domain of this function: {(1, 1), (3, 5), (4, 6), (8, 9)}?
   - **Answer**: a. {1, 3, 4, 8}

3. What is the range of this function: {(1, 0), (2, 0), (−1, 5)}?
   - **Answer**: b. {0, 5}

4. Which of the following relations is a function?
   - **Answer**: b. {(3, 7), (3, 6), (5, 3)}

5. Given f(x) = x − 1, find f(1).
   - **Answer**: a. 0

6. If g(x) = 2x² − 7, what is f(−2)?
   - **Answer**: b. 1

7. Which of the following is the domain of y = 2x + 1?
   - **Answer**: a. {x/x ∈ R}

8. What is the symbol for the inverse of a function?
   - **Answer**: a. f⁻¹(x)

9. It is a function defined by two or more formulas on different parts of its domain.
   - **Answer**: b. Piecewise

10. Which of the following is a piecewise function?
   - **Answer**: d. f(x) = {−18x + 20, x < 19; −16x², x ≥ 19}

11. Evaluate the function f(x) = x² − 3, at x = 3.
   - **Answer**: d. 9

12. If f(x) = x² − x and g(x) = x + 2, what is f(g(x))?
   - **Answer**: c. x² + 4

13. Perform the indicated operation: f(x) = x² − x − 2, g(x) = x + 1. Find f + g(x).
   - **Answer**: c. x² − 3x − 2

14. Use item #13 in finding (f ∘ g)(1).
   - **Answer**: b. 1

15. Given f(x) = x² − 4x + 4, Solve for f(0).
   - **Answer**: d. 4

16. If f(x) = (x − 3)/(x + 4) and g(x) = (x + 1)/(x + 4), find (f − g)(x).
   - **Answer**: c. (x − 4)/(x + 4)

17. Given f(x) = x − 5 and g(x) = x² − 1, Find (f ∘ g)(x).
   - **Answer**: c. x² − 5x − 6

18-19. Answer missing (left for student to solve).

20-21. Answer missing (left for student to solve).

22. Find (g ∘ f)(x) from item number 20.
   - **Answer**: (likely involves composition, answer cannot be confirmed without item 20 info)

23. What is the degree of the polynomial function p(x) = 14 + 4x − 3x³?
   - **Answer**: d. 3

24. Which of the following is a rational function?
   - **Answer**: b. g(x) = 7x³ − 4√x + 1 / x² + 3

25. Which of the following functions is not rational?
   - **Answer**: d. f(x) = (x² − 9) / (x − 3)

26. Which family of functions does y = e^x − 4 belong to?
   - **Answer**: b. Exponential

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Would you like details for any specific questions or further clarifications?

Here are 5 related questions to explore further:
1. What is the difference between a relation and a function?
2. How do you determine the range of a function given its domain?
3. What conditions must a relation satisfy to be classified as a function?
4. How do you find the inverse of a function step by step?
5. What are some real-world applications of piecewise functions?

**Tip**: To verify if a relation is a function, check if each input (x-value) has exactly one output (y-value).

A function is ______ a relation, but not all relations are functions.

This detailed problem set explores key concepts in algebra, including functions, relations, domain and range, and piecewise functions. Solve polynomial and rational function problems suitable for students in Grades 9-12.

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