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Let's go through each part of the question using the given tables for the functions \( f \) and \( g \):

### Tables for Functions \( f \) and \( g \):
- **Function \( f \):**
  - \( f(1) = 2 \)
  - \( f(2) = -3 \)
  - \( f(3) = -1 \)
  - \( f(4) = 3 \)
- **Function \( g \):**
  - \( g(1) = 5 \)
  - \( g(2) = 5 \)
  - \( g(3) = 3 \)
  - \( g(4) = -1 \)

### Solution:
#### (a) Write down the values of \( g(3) \) and \( f^{-1}(3) \):
- **\( g(3) \):** From the table for \( g \), \( g(3) = 3 \).
- **\( f^{-1}(3) \):** To find \( f^{-1}(3) \), we need to find \( x \) such that \( f(x) = 3 \). From the table for \( f \):
  - \( f(4) = 3 \)
  So, \( f^{-1}(3) = 4 \).

#### (b) Calculate \( (f \circ g)(2) \):
- \( (f \circ g)(2) = f(g(2)) \)
- From the table, \( g(2) = 5 \).
- We need \( f(5) \), but the table for \( f \) only gives values for \( x = 1, 2, 3, 4 \). So, we can't calculate \( f(5) \) since it's not given in the table.

#### (c) Calculate \( (g \circ g)(3) \):
- \( (g \circ g)(3) = g(g(3)) \)
- From the table, \( g(3) = 3 \).
- Now, \( g(3) = 3 \) again, so:
  \( g(g(3)) = g(3) = 3 \).

#### (d) Find a solution of the equation \( (g \circ f)(x) = 3 \):
- \( (g \circ f)(x) = g(f(x)) \)
- We need to find \( x \) such that \( g(f(x)) = 3 \).

  - \( f(1) = 2 \) → \( g(f(1)) = g(2) = 5 \) (Not 3)
  - \( f(2) = -3 \) → Not in the table for \( g \)
  - \( f(3) = -1 \) → Not in the table for \( g \)
  - \( f(4) = 3 \) → \( g(f(4)) = g(3) = 3 \) 

  So, \( x = 4 \) is a solution.

### Summary of Solutions:
- (a) \( g(3) = 3 \), \( f^{-1}(3) = 4 \)
- (b) \( (f \circ g)(2) \): Cannot be determined from the given information.
- (c) \( (g \circ g)(3) = 3 \)
- (d) \( x = 4 \)

Would you like further details or have any questions about these calculations?

Here are some related questions to explore:
1. How do you find the inverse of a function using its table of values?
2. What does the composition of functions represent?
3. How would you determine \( (f \circ g)(x) \) if the tables provided values for all integers?
4. What if \( g \) had a different range? How would it affect finding \( (g \circ f)(x) = 3 \)?
5. How can you use function composition in real-life problem solving?

**Tip:** When working with tables of values for functions, always ensure that the function's domain and range fit within the given data to avoid undefined situations.

The tables below show some values of two functions f and g. Write down the values of g(3) and f^{-1}(3), Calculate (f ∘ g)(2), Calculate (g ∘ g)(3), Find a solution of the equation (g ∘ f)(x) = 3.

Learn how to solve function composition and inverse function problems using tables of values for functions f and g. This exercise covers concepts such as g(3), f^{-1}(3), and (g ∘ f)(x) = 3. Suitable for students in Grades 9-11.

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