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The image shows a math exercise that involves identifying the type of each function and finding its domain and range. The functions are labeled 1 to 7. Let's break down each function, identify its type, and determine the domain and range:

### 1. \( f(x) = x^2 - 1 \)
- **Type**: Quadratic function (since it's a polynomial of degree 2).
- **Domain**: All real numbers \( \mathbb{R} \) (since any real number can be squared).
- **Range**: The function has a minimum value of \(-1\) (at \( x = 0 \)) and increases to infinity. So, \( \text{Range} = [-1, \infty) \).

### 2. \( f(x) = \sqrt{x - 2} \)
- **Type**: Square root function.
- **Domain**: \( x - 2 \geq 0 \), meaning \( x \geq 2 \). Thus, \( \text{Domain} = [2, \infty) \).
- **Range**: The square root function produces only non-negative outputs, so the range is \( [0, \infty) \).

### 3. \( f(x) = \sqrt{x^2 - 2} \)
- **Type**: Square root of a quadratic function.
- **Domain**: \( x^2 - 2 \geq 0 \), which simplifies to \( x \leq -\sqrt{2} \) or \( x \geq \sqrt{2} \). So, \( \text{Domain} = (-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty) \).
- **Range**: The range starts at 0 and increases to infinity. Therefore, \( \text{Range} = [0, \infty) \).

### 4. \( f(x) = \dfrac{1}{x - 2} \)
- **Type**: Rational function.
- **Domain**: The function is undefined when the denominator is zero, i.e., \( x \neq 2 \). Thus, \( \text{Domain} = (-\infty, 2) \cup (2, \infty) \).
- **Range**: The function's values can approach both negative and positive infinity but never reach 0, so \( \text{Range} = (-\infty, 0) \cup (0, \infty) \).

### 5. \( f(x) = \dfrac{1}{\sqrt{x}} \)
- **Type**: Rational function with a square root.
- **Domain**: The square root is defined for \( x \geq 0 \), and the denominator cannot be zero, so \( x > 0 \). Thus, \( \text{Domain} = (0, \infty) \).
- **Range**: The function approaches infinity as \( x \to 0^+ \) and decreases towards 0 as \( x \to \infty \). So, \( \text{Range} = (0, \infty) \).

### 6. \( f(x) = \sqrt{4 - x^2} \)
- **Type**: Square root of a quadratic function (circular segment).
- **Domain**: \( 4 - x^2 \geq 0 \), so \( -2 \leq x \leq 2 \). Thus, \( \text{Domain} = [-2, 2] \).
- **Range**: The function produces values between 0 and 2, inclusive, so \( \text{Range} = [0, 2] \).

### 7. \( f(x) = |x - 1| + 1 \)
- **Type**: Absolute value function.
- **Domain**: The absolute value function is defined for all real numbers, so \( \text{Domain} = \mathbb{R} \).
- **Range**: The minimum value of the function is 1 (when \( x = 1 \)), and it increases without bound. Thus, \( \text{Range} = [1, \infty) \).

Would you like further clarification on any of these functions?

### Additional Questions:
1. How do you graph a quadratic function like \( f(x) = x^2 - 1 \)?
2. Can you explain how to solve for the domain of functions involving square roots?
3. What is the relationship between the degree of a polynomial and the shape of its graph?
4. How does the range of a rational function differ from that of a quadratic function?
5. What are some key differences between absolute value functions and quadratic functions?

### Tip:
When finding the domain of a function, always check for values that make the expression inside a square root negative or the denominator equal to zero, as these create restrictions.

Determine the type of each function, sketch the graph, and find its domain & range for the following functions: f(x) = x^2 - 1, f(x) = sqrt(x - 2), f(x) = sqrt(x^2 - 2), f(x) = 1 / (x - 2), f(x) = 1 / sqrt(x), f(x) = sqrt(4 - x^2), f(x) = |x - 1| + 1.

This exercise breaks down various function types including quadratic, square root, and rational functions. The goal is to identify the type of function, determine its domain, and range for each case. Suitable for Grades 10-12.

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