Let's analyze and solve the questions from the image you've provided.

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.