Unit 3 Homework 1 Relations Domain Range And Functions

Let's delve into the foundational concepts of relations, domains, ranges, and functions. These building blocks are essential for understanding more advanced mathematical topics. Through clear explanations and practical examples, we'll demystify these terms and show how they intertwine.

Relations: Connecting the Dots

A relation is simply a set of ordered pairs. An ordered pair consists of two elements, often denoted as (x, y), where the order matters. In the context of relations, 'x' and 'y' can represent numbers, objects, or any type of element.

Think of a relation as a way to describe a connection or correspondence between elements from two sets. These sets don't have to be distinct; you can have a relation mapping elements from a set to itself.

Representing Relations:

Relations can be represented in several ways:

Set of Ordered Pairs: The most fundamental representation is listing the ordered pairs within curly braces. For example: {(1, 2), (3, 4), (5, 6)} is a relation.
Table: You can organize the ordered pairs into a table with columns representing the 'x' and 'y' values.

x y

1 2

3 4

5 6
Graph: Each ordered pair can be plotted as a point on a coordinate plane (Cartesian plane). The relation is then visually represented by the collection of these points.
Mapping Diagram: This diagram shows two sets with arrows connecting elements from the first set to related elements in the second set.
Equation: Sometimes, a relation can be defined by an equation that relates 'x' and 'y'. For instance, y = x + 1 defines a relation. Any ordered pair (x, y) that satisfies this equation belongs to the relation.

x	y
1	2
3	4
5	6

Examples of Relations:

Student-Grade Relation: Suppose you have a class of students and their corresponding grades. The relation could be represented as {(Alice, A), (Bob, B), (Charlie, C)}, indicating Alice got an A, Bob got a B, and Charlie got a C.
Number Relation: Consider the relation "is less than". We can express this relation between the numbers 1, 2, and 3 as {(1, 2), (1, 3), (2, 3)}.
Geometric Relation: Think of points on a circle. The relation could be defined as all points (x, y) that satisfy the equation x² + y² = r², where 'r' is the radius of the circle.

Domain: Where the Relation Begins

The domain of a relation is the set of all possible first elements (usually the 'x' values) in the ordered pairs. It's essentially the set of inputs that the relation accepts.

Finding the Domain:

Set of Ordered Pairs: Simply collect all the 'x' values from the ordered pairs. If there are duplicates, list the element only once in the domain. For example, if the relation is {(1, 2), (3, 4), (1, 5)}, the domain is {1, 3}.
Table: The domain is the set of all values in the first column (the 'x' column).
Graph: Project all the points of the relation onto the x-axis. The domain is the set of all x-values that are covered by these projections.
Equation: Determining the domain from an equation requires considering any restrictions on the values of 'x'. These restrictions might arise from:
- Division by zero: The denominator of a fraction cannot be zero. Therefore, any value of 'x' that makes the denominator zero is excluded from the domain.
- Square roots (or other even roots) of negative numbers: In the realm of real numbers, you can't take the square root of a negative number. So, the expression under the square root must be greater than or equal to zero.
- Logarithms of non-positive numbers: The argument of a logarithm (the expression inside the log) must be strictly positive.

Examples of Domains:

Relation: {(2, 4), (5, 7), (8, 10)}. Domain: {2, 5, 8}.
Relation defined by the equation: y = 1/x. Domain: All real numbers except x = 0 (because division by zero is undefined). This can be written as: {x | x ∈ ℝ, x ≠ 0} or (-∞, 0) ∪ (0, ∞).
Relation defined by the equation: y = √x. Domain: All non-negative real numbers. This can be written as: {x | x ∈ ℝ, x ≥ 0} or [0, ∞).

Range: Where the Relation Ends

The range of a relation is the set of all possible second elements (usually the 'y' values) in the ordered pairs. It's the set of all outputs that the relation produces.

Finding the Range:

Set of Ordered Pairs: Collect all the 'y' values from the ordered pairs, removing any duplicates. For instance, the relation {(1, 2), (3, 2), (4, 5)} has a range of {2, 5}.
Table: The range is the set of all values in the second column (the 'y' column).
Graph: Project all the points of the relation onto the y-axis. The range is the set of all y-values that are covered by these projections.
Equation: Determining the range from an equation can be more challenging than finding the domain. It often involves:
- Analyzing the behavior of the equation as 'x' varies over its domain.
- Identifying any maximum or minimum values that 'y' can attain.
- Considering any asymptotes (lines that the graph approaches but never touches).

Examples of Ranges:

Relation: {(1, 3), (2, 6), (3, 9)}. Range: {3, 6, 9}.
Relation defined by the equation: y = x². Range: All non-negative real numbers because squaring any real number results in a non-negative value. This can be written as: {y | y ∈ ℝ, y ≥ 0} or [0, ∞).
Relation defined by the equation: y = sin(x). Range: [-1, 1] because the sine function always produces values between -1 and 1, inclusive.

Functions: A Special Type of Relation

A function is a special type of relation where each element in the domain is paired with exactly one element in the range. In other words, for every input 'x', there is only one output 'y'. This is often described as the "vertical line test" when looking at a graph.

Key Properties of Functions:

Uniqueness of Output: For each 'x' in the domain, there is only one corresponding 'y' value.
Every Input Must Be Mapped: Every element in the domain must be associated with an element in the range. There cannot be any "unassigned" inputs.

The Vertical Line Test:

A visual way to determine if a graph represents a function is the vertical line test. If you can draw any vertical line that intersects the graph at more than one point, then the graph does not represent a function. This is because a vertical line represents a single 'x' value, and if it intersects the graph at multiple points, it means that single 'x' value is associated with multiple 'y' values, violating the uniqueness of output requirement.

Function Notation:

Functions are often denoted using function notation:

f(x) represents the value of the function f at the input x.
For example, if f(x) = x² + 1, then f(2) = 2² + 1 = 5. This means that when the input is 2, the output of the function f is 5.

Examples of Functions and Non-Functions:

Function: f(x) = 2x + 3. This is a function because for every 'x' value, there is only one corresponding 'y' value.
Function: {(1, 2), (2, 4), (3, 6), (4, 8)}. This is a function because each 'x' value is paired with only one 'y' value.
Not a Function: {(1, 2), (1, 3), (2, 4)}. This is not a function because the 'x' value 1 is paired with two different 'y' values (2 and 3).
Not a Function: x² + y² = 1 (equation of a circle). This is not a function. If you solve for 'y', you get y = ±√(1 - x²), which means that for many 'x' values, there are two corresponding 'y' values (one positive and one negative). Also, it fails the vertical line test.

Determining Domain and Range from Equations: More Examples

Let's explore more examples to solidify your understanding of finding the domain and range from equations.

Example 1: Rational Function

Consider the function f(x) = (x + 2) / (x - 3).

Domain: The denominator cannot be zero, so x - 3 ≠ 0, which means x ≠ 3. Therefore, the domain is all real numbers except 3: {x | x ∈ ℝ, x ≠ 3} or (-∞, 3) ∪ (3, ∞).
Range: To find the range, we can try to solve the equation for x in terms of y:
1. y = (x + 2) / (x - 3)
2. y(x - 3) = x + 2
3. yx - 3y = x + 2
4. yx - x = 3y + 2
5. x(y - 1) = 3y + 2
6. x = (3y + 2) / (y - 1)
Now, we see that the denominator y - 1 cannot be zero, so y ≠ 1. Therefore, the range is all real numbers except 1: {y | y ∈ ℝ, y ≠ 1} or (-∞, 1) ∪ (1, ∞).

Example 2: Square Root Function

Consider the function g(x) = √(4 - x²).

Domain: The expression under the square root must be non-negative, so 4 - x² ≥ 0. This implies x² ≤ 4, which means -2 ≤ x ≤ 2. Therefore, the domain is [-2, 2].
Range: The square root function always returns non-negative values. The maximum value of g(x) occurs when x = 0, and g(0) = √4 = 2. The minimum value is 0, which occurs when x = ±2. Therefore, the range is [0, 2].

Example 3: Absolute Value Function

Consider the function h(x) = |x - 1| + 2.

Domain: The absolute value function is defined for all real numbers, so there are no restrictions on x. Therefore, the domain is all real numbers: {x | x ∈ ℝ} or (-∞, ∞).
Range: The absolute value |x - 1| is always non-negative, so its minimum value is 0, which occurs when x = 1. Therefore, the minimum value of h(x) is 0 + 2 = 2. Since |x - 1| can be arbitrarily large, h(x) can also be arbitrarily large. Therefore, the range is [2, ∞).

Example 4: Polynomial Function

Consider the function p(x) = x³ - 2x + 1.

Domain: Polynomial functions are defined for all real numbers. Therefore, the domain is all real numbers: {x | x ∈ ℝ} or (-∞, ∞).
Range: Cubic functions (polynomials of degree 3) have a range of all real numbers. This is because as x approaches positive infinity, p(x) also approaches positive infinity, and as x approaches negative infinity, p(x) also approaches negative infinity. Therefore, the range is all real numbers: {y | y ∈ ℝ} or (-∞, ∞).

Common Mistakes to Avoid

Confusing Domain and Range: Always remember that the domain refers to the input values (usually 'x'), and the range refers to the output values (usually 'y').
Forgetting Restrictions: Pay close attention to potential restrictions on the domain, such as division by zero, square roots of negative numbers, and logarithms of non-positive numbers.
Assuming the Range is Obvious: Determining the range can be tricky. Carefully analyze the function's behavior to identify any minimum or maximum values, asymptotes, or other limitations on the possible output values.
Incorrectly Applying the Vertical Line Test: Make sure you understand that the vertical line test only applies to graphs. It cannot be used to determine if a relation is a function based on its equation or a set of ordered pairs.
Not Simplifying: Always simplify expressions as much as possible before determining the domain or range. This can help you identify restrictions and understand the function's behavior more easily.

Practice Problems

Test your understanding with these practice problems:

Find the domain and range of the relation {(0, 1), (1, 3), (2, 5), (3, 7)}. Is it a function?
Find the domain and range of the function f(x) = √(x + 5).
Find the domain and range of the function g(x) = 1 / (x² - 4).
Determine if the relation defined by the equation x = y² is a function.
Sketch the graph of y = |x|. What is the domain and range? Is it a function?

Conclusion

Understanding relations, domains, ranges, and functions is paramount for mastering mathematics. By mastering these concepts, you lay a solid foundation for more advanced topics such as calculus, linear algebra, and differential equations. Remember to practice, apply the concepts to diverse examples, and pay attention to potential pitfalls. With consistent effort, you will confidently navigate the world of mathematical relationships.