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Let's go through the questions one by one.

### Question 1:
For \( f(x) = 4x^2 - 3x + 1 \), evaluate:
1. \( f(a) \)
2. \( f(a + h) \)
3. \( f(a + h) - f(a) \)

- **\( f(a) \):**
  Substitute \( x = a \) into the function:
  \[
  f(a) = 4a^2 - 3a + 1
  \]

- **\( f(a + h) \):**
  Substitute \( x = a + h \):
  \[
  f(a + h) = 4(a + h)^2 - 3(a + h) + 1
  \]
  Expand the terms:
  \[
  = 4(a^2 + 2ah + h^2) - 3(a + h) + 1
  = 4a^2 + 8ah + 4h^2 - 3a - 3h + 1
  \]

- **\( f(a + h) - f(a) \):**
  Subtract \( f(a) \) from \( f(a + h) \):
  \[
  f(a + h) - f(a) = \left(4a^2 + 8ah + 4h^2 - 3a - 3h + 1\right) - \left(4a^2 - 3a + 1\right)
  \]
  Simplifying:
  \[
  = 8ah + 4h^2 - 3h
  \]

### Question 2:
For \( f(x) = \frac{1}{x} \), evaluate:
1. \( f(a) \)
2. \( f(a + h) \)
3. \( f(a + h) - f(a) \)

- **\( f(a) \):**
  \[
  f(a) = \frac{1}{a}
  \]

- **\( f(a + h) \):**
  \[
  f(a + h) = \frac{1}{a + h}
  \]

- **\( f(a + h) - f(a) \):**
  \[
  f(a + h) - f(a) = \frac{1}{a + h} - \frac{1}{a}
  \]
  Use a common denominator to simplify:
  \[
  f(a + h) - f(a) = \frac{a - (a + h)}{a(a + h)} = \frac{-h}{a(a + h)}
  \]

### Question 3:
For \( f(x) = \sqrt{3 - 2x} \), find the:
1. **x-intercept**: Set \( f(x) = 0 \), which gives:
   \[
   \sqrt{3 - 2x} = 0 \implies 3 - 2x = 0 \implies x = \frac{3}{2}
   \]
   So, the x-intercept is \( \left(\frac{3}{2}, 0\right) \).

2. **y-intercept**: Set \( x = 0 \), and evaluate \( f(0) \):
   \[
   f(0) = \sqrt{3 - 2(0)} = \sqrt{3}
   \]
   So, the y-intercept is \( (0, \sqrt{3}) \).

### Question 4:
Find the domain and range for \( y = \sqrt{8x - 1} - 1 \).

- **Domain**: The expression inside the square root must be non-negative:
  \[
  8x - 1 \geq 0 \implies x \geq \frac{1}{8}
  \]
  So, the domain is \( \left[\frac{1}{8}, \infty \right) \).

- **Range**: The function \( \sqrt{8x - 1} \) has a minimum value of 0, so the range of \( y = \sqrt{8x - 1} - 1 \) is:
  \[
  y \geq -1
  \]
  Therefore, the range is \( [-1, \infty) \).

### Questions 5 to 8:
Would you like me to continue with the rest of the questions or clarify anything from the above steps?

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### Related Questions:
1. How do we evaluate functions at \( f(a+h) \) for more complex functions?
2. How do we find the domain and range for functions involving fractions?
3. What are the conditions for a function to have an x-intercept?
4. Can you explain why square roots impact the domain of functions?
5. How would the x-intercept calculation differ if the function were \( \sqrt{x^2 - 4x} \)?

### Tip:
Always check the domain of a function before performing further calculations, especially when dealing with square roots and fractions!

Assignment 1 questions: The user submitted a set of 8 math problems covering functions, domain and range, and intercepts.

Solve a series of algebra problems involving quadratic functions, domain and range analysis, intercepts, and square root functions. Detailed step-by-step solutions for students in Grades 10-12, covering core mathematical concepts and formulas.

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