Let's determine the domain and range for each function.

a. $\{(1,2), (3,5), (6,7)\}$

This is a set of ordered pairs.

• Domain: The domain is the set of all first elements (x-values) in the ordered pairs. $\text{Domain} = \{1, 3, 6\}$

• Range: The range is the set of all second elements (y-values) in the ordered pairs. $\text{Range} = \{2, 5, 7\}$

b. $f(x) = 7$

This is a constant function, meaning for every input $x$, the output is always 7.

• Domain: Since there are no restrictions on $x$, the domain is all real numbers. $\text{Domain} = \mathbb{R}$

• Range: The range is the single value that $f(x)$ outputs, which is 7. $\text{Range} = \{7\}$

c. $f(x) = 2x + 9$

This is a linear function.

• Domain: A linear function has no restrictions, so the domain is all real numbers. $\text{Domain} = \mathbb{R}$

• Range: Since the slope $2$ is non-zero, the function can take any real value, so the range is all real numbers. $\text{Range} = \mathbb{R}$

d. $f(x) = x^2 - 2x + 1$

This is a quadratic function, which opens upwards because the coefficient of $x^2$ is positive.

• Domain: The domain of a quadratic function is all real numbers. $\text{Domain} = \mathbb{R}$

• Range: To find the range, we can complete the square or find the vertex of the parabola. The function can be rewritten as: $f(x) = (x-1)^2$ The vertex is at $x = 1$, and the minimum value is $0$. Thus, the range is all real numbers greater than or equal to 0. $\text{Range} = [0, \infty)$

e. $f(x) = -x^2 + 2x + 2$

This is a quadratic function that opens downwards because the coefficient of $x^2$ is negative.

• Domain: The domain of a quadratic function is all real numbers. $\text{Domain} = \mathbb{R}$

• Range: We can find the range by determining the vertex, which will give the maximum value of the function. Rewriting the function by completing the square: $f(x) = -(x^2 - 2x) + 2 = -(x-1)^2 + 3$ The vertex is at $x = 1$, and the maximum value is $3$. Therefore, the range is all real numbers less than or equal to 3. $\text{Range} = (-\infty, 3]$

If you need more details or have any questions, feel free to ask! Here are some related questions:

1. How do you determine the domain of a rational function?
2. What is the significance of the vertex in a quadratic function?
3. How do transformations affect the domain and range of functions?
4. How can the range of a quadratic function be determined using calculus?
5. What are the domain and range of the absolute value function $f(x) = |x|$?

Tip: For quadratic functions, completing the square is a reliable method to find the vertex, which helps in determining the range.