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Let's analyze the problem step by step.

You are given two functions:
- \( f(x) = 2x^2 - 1 \)
- \( g(x) = x^2 + 1 \)

Now, let's evaluate the statements one by one:

### I. \( f(g(2)) = 49 \)
- First, calculate \( g(2) \):
  \[
  g(2) = 2^2 + 1 = 4 + 1 = 5
  \]
- Now, calculate \( f(5) \):
  \[
  f(5) = 2(5^2) - 1 = 2(25) - 1 = 50 - 1 = 49
  \]
- So, \( f(g(2)) = 49 \), which is **True**.

### II. \( g(f(2)) = 50 \)
- First, calculate \( f(2) \):
  \[
  f(2) = 2(2^2) - 1 = 2(4) - 1 = 8 - 1 = 7
  \]
- Now, calculate \( g(7) \):
  \[
  g(7) = 7^2 + 1 = 49 + 1 = 50
  \]
- So, \( g(f(2)) = 50 \), which is **True**.

### III. \( f(4) = 30 \)
- Calculate \( f(4) \):
  \[
  f(4) = 2(4^2) - 1 = 2(16) - 1 = 32 - 1 = 31
  \]
- So, \( f(4) = 31 \), not 30. This is **False**.

### IV. \( g(4) = 17 \)
- Calculate \( g(4) \):
  \[
  g(4) = 4^2 + 1 = 16 + 1 = 17
  \]
- So, \( g(4) = 17 \), which is **True**.

### Final Evaluation:
- I: True
- II: True
- III: False
- IV: True

The correct answer is **B: V-V-F-V**.

Would you like further details or have any questions?

### Relative Questions:
1. How do we interpret the composition of two functions in mathematics?
2. What are the general steps for solving a function composition problem?
3. Can you provide examples where function compositions are used in real-world applications?
4. What is the difference between evaluating a single function and a composition of functions?
5. How does the quadratic form in both functions impact the results?

### Tip:
When evaluating composed functions, always compute the innermost function first before proceeding to the outer function.

De acordo com as seguintes funções f(x) = 2x^2 - 1 e g(x) = x^2 + 1, analise os resultados abaixo e classifique como verdadeira (V) ou falsa (F).
I. f(g(2)) = 49
II. g(f(2)) = 50
III. f(4) = 30
IV. g(4) = 17

Analyze and classify as true or false the results of function composition using the quadratic functions f(x) = 2x^2 - 1 and g(x) = x^2 + 1. Suitable for high school students, this problem covers core concepts of function composition.

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