I can see the mathematical questions in the image you uploaded. Let me break down the tasks provided:

I. Compute the difference quotient for the given function:

The difference quotient is defined as:

$\frac{f(x+h) - f(x)}{h}$

You need to compute this quotient for the following functions:

1. $f(x) = 8x - 1$
2. $f(x) = 3x^2$
3. $f(x) = 7 - x^2$
4. $f(x) = 3x^2 + 7x - 4$
5. $f(x) = \frac{2}{x}$

II. Determine the domain of the function using interval notation:

For these functions, you need to specify the domain:

6. $A(x) = 6x + 14$
7. $f(x) = \frac{1}{x^2 - 25}$
8. $g(t) = \frac{8t - 24}{t^2 - 7t - 18}$
9. $g(w) = \sqrt{9w + 7}$
10. $f(x) = \frac{1}{\sqrt{x^2 - 8x + 15}}$

III. Find the following compositions of functions:

You are asked to work with composite functions in the following way:

a. Find $f \circ g \circ h$, where:

• $f(x) = \sqrt{1 - x}$
• $g(x) = 1 - x^2$
• $h(x) = 1 + \sqrt{x}$

b. If $T(x) = \frac{1}{1 + \sqrt{x}}$, find functions $f$, $g$, and $h$ such that $f \circ g \circ h = T$.

Would you like me to solve each of these problems step by step? If so, please let me know which part you'd like to start with, or if you'd like to go through all of them in order.

Example follow-up questions:

1. How do you simplify the difference quotient for quadratic functions?
2. What are the steps for finding the domain of a rational function?
3. How do you compute composite functions for complex expressions?
4. What special considerations are there for square root functions in domains?
5. How do you decompose functions for compositions like in part III?

Tip:

When computing the difference quotient for functions with fractions, be mindful of handling the denominators carefully to avoid mistakes.