The image contains mathematical functions, and the task is to determine the domain and range for each of these functions. The functions listed are:

a. $f(x) = 3x - 8$

b. $f(x) = \sqrt{7x - 3}$

c. $f(x) = \frac{x-2}{x+7}$

d. $f(x) = \sqrt{4x - 6}$

e. $f(x) = \frac{3x - 7}{x^2 + 6x + 5}$

Let's analyze each function to determine their respective domains and ranges:

a. $f(x) = 3x - 8$

• Domain: All real numbers, since there's no restriction on $x$. $\mathbb{R}$
• Range: Since this is a linear function, the range is also all real numbers. $\mathbb{R}$

b. $f(x) = \sqrt{7x - 3}$

• Domain: The expression inside the square root must be non-negative. $7x - 3 \geq 0 \Rightarrow x \geq \frac{3}{7}$ So, the domain is $\left[\frac{3}{7}, \infty\right)$.
• Range: The output of a square root function is always non-negative, so the range is $[0, \infty)$.

c. $f(x) = \frac{x-2}{x+7}$

• Domain: The denominator cannot be zero, so $x \neq -7$. Thus, the domain is $\mathbb{R} \setminus \{-7\}$.
• Range: This is a rational function, so the range is all real numbers except the horizontal asymptote. For this function, as $x \to \infty$ or $x \to -\infty$, $f(x)$ approaches 1. Hence, the range is $\mathbb{R} \setminus \{1\}$.

d. $f(x) = \sqrt{4x - 6}$

• Domain: The expression inside the square root must be non-negative. $4x - 6 \geq 0 \Rightarrow x \geq \frac{3}{2}$ So, the domain is $\left[\frac{3}{2}, \infty\right)$.
• Range: The output of a square root function is always non-negative, so the range is $[0, \infty)$.

e. $f(x) = \frac{3x - 7}{x^2 + 6x + 5}$

• Domain: The denominator cannot be zero, so we need to solve $x^2 + 6x + 5 = 0$. Factorizing: $(x + 5)(x + 1) = 0$ So, $x = -5$ and $x = -1$ make the denominator zero. Thus, the domain is $\mathbb{R} \setminus \{-5, -1\}$.
• Range: The range is more complex to determine exactly, but it's typically all real numbers except where the numerator causes the function to approach infinity or the function simplifies to a constant value. In this case, it's all real numbers except where $f(x) = 0$ (since it has no asymptote at $y = 0$).

Would you like further details on any of these steps, or do you have any questions? Here are a few related problems you could consider:

1. What is the domain and range of the function $f(x) = \frac{1}{x}$?
2. How would you find the domain and range for $f(x) = \sqrt{x^2 - 4x + 4}$?
3. Can a rational function ever have a range that is all real numbers?
4. How do you determine if a function has a horizontal asymptote?
5. How does the range of a quadratic function differ from that of a linear function?

Tip: When determining the range of more complex functions, consider solving $y = f(x)$ for $x$ to express $x$ in terms of $y$ and then determine which values of $y$ are possible.