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### Question 7
The problem asks for the value of \((g \circ f)(1)\), where \( (g \circ f)(x) = g(f(x)) \).

Given:
- \( f(1) = -1 \)
- \( g(-1) = 4 \)

We calculate:
1. First, find \( f(1) \): \( f(1) = -1 \).
2. Then, substitute this into \( g(x) \): \( g(-1) = 4 \).

So, \((g \circ f)(1) = g(f(1)) = g(-1) = 4\).

The statement says that the value is \(-1\), but we found it to be \(4\), so the correct answer is **False**.

### Question 8
The problem asks whether the functions \( f(x) = 3x + 4 \) and \( g(x) = \frac{x - 4}{3} \) are inverses of each other.

To verify, compose the two functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \):

1. **\( f(g(x)) \)**:
   \[
   f(g(x)) = f\left(\frac{x - 4}{3}\right) = 3\left(\frac{x - 4}{3}\right) + 4 = x - 4 + 4 = x
   \]

2. **\( g(f(x)) \)**:
   \[
   g(f(x)) = g(3x + 4) = \frac{3x + 4 - 4}{3} = \frac{3x}{3} = x
   \]

Since both compositions \( (f \circ g)(x) \) and \( (g \circ f)(x) \) equal \( x \), the functions are indeed inverses. So, the correct answer is **True**.

### Question 9
The problem asks for the domain of the composition \( D_{g \circ f} \) where:
- \( f(x) = x^2 - 2x \)
- \( g(x) = x^2 + 1 \)

The domain of the composition \( g(f(x)) \) is determined by the domain of \( f(x) \) and the range of \( f(x) \).

1. The domain of \( f(x) \), which is a quadratic function, is all real numbers \( (-\infty, \infty) \).
2. The range of \( f(x) \) is also all real numbers because it's a quadratic function, which opens upwards, with the minimum at \( f(1) = -1 \).

Since \( g(x) = x^2 + 1 \) also has a domain of all real numbers \( (-\infty, \infty) \), and since \( g(x) \) is defined for all \( f(x) \), the domain \( D_{g \circ f} \) will also be all real numbers.

Thus, the correct answer is **True**.

Would you like to go into more details or ask any further questions? 

Here are some relative questions:
1. What is the significance of the composition of functions in mathematics?
2. Can you explain how to find the inverse of a function?
3. What are the properties of quadratic functions that determine their domain and range?
4. How do you determine if a given function is injective (one-to-one)?
5. How does the domain and range of a function affect the composition of two functions?

**Tip**: Always check both the domain and range when dealing with function compositions, as they directly affect the possible inputs and outputs.

The table contains some input-output info for the functions f and g. The value of (g∘f)(1) = -1. The functions f(x) = 3x + 4 and g(x) = (x - 4)/3 are inverse of each other. If f(x) = x² - 2x and g(x) = x² + 1, then Dg∘f is (-∞, ∞).

Learn how to compute the composition of two functions, verify if two functions are inverses, and determine the domain of a composition. This problem involves function composition (g∘f), inverse functions, and domain analysis, suitable for Grades 9-12.

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