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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.

Explore the domain and range of mathematical functions including linear, square root, and rational functions. Understand the conditions and calculations required to determine these ranges and domains effectively. Suitable for Grades 10-12.

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