https://file.aiquickdraw.com/mathbot/file/17279959510369p63xfuj.jpg

Let's go through each part of the problem based on the image.

We have two functions \( f(x) \) and \( g(x) \) given in tabular form:

- \( f(x) \):
  - \( f(1) = 4 \)
  - \( f(2) = 1 \)
  - \( f(3) = 5 \)
  - \( f(4) = 5 \)
  - \( f(5) = 3 \)

- \( g(x) \):
  - \( g(1) = 4 \)
  - \( g(2) = 4 \)
  - \( g(3) = 2 \)
  - \( g(4) = 2 \)
  - \( g(5) = 3 \)

### (a) What is \( f(3) \)?
From the table for \( f(x) \), we see that \( f(3) = 5 \).

### (b) Compute \( (f \circ g)(5) \), i.e., \( f(g(5)) \).
- First, find \( g(5) \): From the table for \( g(x) \), we know that \( g(5) = 3 \).
- Now, substitute into \( f(x) \): \( f(g(5)) = f(3) \).
- From part (a), \( f(3) = 5 \).

Thus, \( (f \circ g)(5) = 5 \).

### (c) Compute \( (g \circ f)(3) \), i.e., \( g(f(3)) \).
- First, find \( f(3) \): From part (a), \( f(3) = 5 \).
- Now, substitute into \( g(x) \): \( g(f(3)) = g(5) \).
- From the table for \( g(x) \), \( g(5) = 3 \).

Thus, \( (g \circ f)(3) = 3 \).

### (d) Compute \( (f^{-1} \circ g)(2) \), i.e., \( f^{-1}(g(2)) \).
- First, find \( g(2) \): From the table for \( g(x) \), \( g(2) = 4 \).
- Now, find \( f^{-1}(4) \). This means we are looking for the value of \( x \) such that \( f(x) = 4 \). From the table for \( f(x) \), \( f(1) = 4 \), so \( f^{-1}(4) = 1 \).

Thus, \( (f^{-1} \circ g)(2) = 1 \).

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Would you like more details or clarifications on any of the steps? 

Here are 5 related questions for practice:
1. What is \( g(f(4)) \)?
2. Compute \( (f \circ g)(3) \).
3. Find \( f^{-1}(5) \).
4. What is \( (g \circ g)(1) \)?
5. Compute \( f(g(4)) \).

**Tip:** For composite functions like \( (f \circ g)(x) \), always evaluate the inside function first, then apply the outside function!

What is f(3)? Compute (f ∘ g)(5), (g ∘ f)(3), and (f⁻¹ ∘ g)(2) using the given functions f(x) and g(x) from a table.

Explore function composition and inverse functions through a step-by-step solution to a problem involving f(x) and g(x) given in tabular form. Learn how to compute f(3), (f ∘ g)(5), (g ∘ f)(3), and (f⁻¹ ∘ g)(2). Suitable for high school students in Grades 10-12.

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