Determine the domain and range of the following functions.

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

b.f(x) = 7

c.f(x) = 2x+9

d.f(x) = x2 – 2x + 1

e.f(x)= -x² + 2x + 2 

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.

Explore how to determine the domain and range of various functions including constant, linear, and quadratic functions with detailed examples and solutions. Learn about quadratic functions, their vertex form, and the method of completing the square. Suitable for Grades 9-12 students and anyone interested in understanding functions and their properties.

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