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.