Unit 3 Test Study Guide Relations And Functions Answer Key

Let's delve into the world of relations and functions, providing a comprehensive study guide that not only prepares you for your Unit 3 test but also equips you with a solid understanding of the underlying concepts. This guide will cover key definitions, properties, and examples, ensuring you're ready to tackle any question that comes your way.

Relations and Functions: A Comprehensive Study Guide

Understanding relations and functions is fundamental in mathematics. They form the building blocks for more advanced topics and are crucial for solving various problems in algebra, calculus, and beyond. This guide will break down the core concepts, explore different representations, and offer practice questions to solidify your knowledge.

What are Relations?

A relation is simply a set of ordered pairs. An ordered pair is a combination of two elements, typically written as (x, y), where the order matters. The set of all first elements (x-values) in the ordered pairs is called the domain of the relation, while the set of all second elements (y-values) is called the range of the relation.

Representing Relations:

Relations can be represented in several ways:

Set of Ordered Pairs: This is the most basic representation, listing all the ordered pairs in a set. For example: {(1, 2), (3, 4), (5, 6)}
Table: A table organizes the ordered pairs into rows and columns, with one column for the x-values and another for the y-values.

x y

1 2

3 4

5 6
Graph: The ordered pairs can be plotted as points on a coordinate plane, visually representing the relation.
Mapping Diagram: A mapping diagram uses arrows to connect elements from the domain to corresponding elements in the range.
Equation/Rule: Sometimes, a relation can be defined by an equation that relates x and y. For instance, y = 2x + 1.

x	y
1	2
3	4
5	6

Examples of Relations:

{(2, 4), (3, 9), (4, 16), (5, 25)}: This relation represents the squares of numbers. The domain is {2, 3, 4, 5}, and the range is {4, 9, 16, 25}.
{(-1, 1), (0, 0), (1, 1), (2, 4)}: This relation could also represent squares, or part of a quadratic function. The domain is {-1, 0, 1, 2}, and the range is {0, 1, 4}.
y = x^2 + 1: This equation defines a relation where the y-value is always one more than the square of the x-value.

What are Functions?

A function is a special type of relation. It's a relation where each element in the domain (x-value) is paired with exactly one element in the range (y-value). In other words, for every input (x), there is only one output (y).

Key Characteristics of a Function:

Unique Output: Each input has a single, unique output.
Vertical Line Test: Graphically, a relation is a function if and only if no vertical line intersects the graph more than once. If a vertical line intersects the graph at two or more points, it means that one x-value is associated with multiple y-values, and therefore, it's not a function.

Examples of Functions:

y = 3x + 2: This is a linear function. For any x-value you input, you'll get a single, unique y-value.
f(x) = x^3 - 5: This is a cubic function. Again, each x-value will produce only one y-value.
{(1, 5), (2, 10), (3, 15), (4, 20)}: This set of ordered pairs represents a function because each x-value (1, 2, 3, 4) is associated with only one y-value (5, 10, 15, 20) respectively.

Examples of Relations that are NOT Functions:

{(1, 2), (1, 3), (2, 4), (3, 5)}: This is not a function because the x-value '1' is associated with two different y-values (2 and 3).
x = y^2: This is not a function. For example, if x = 4, then y could be 2 or -2. Therefore, one x-value has two corresponding y-values.
A circle centered at the origin. Any vertical line drawn through the circle (except those tangent to the leftmost and rightmost points) will intersect the circle at two points, meaning it fails the vertical line test.

Function Notation:

Function notation is a standard way of writing functions. Instead of writing y = ..., we use f(x) = .... Here, f is the name of the function, and x is the input variable. f(x) represents the output value of the function for a given input x.

f(x) = 2x + 3: This means the function f takes an input x, multiplies it by 2, and then adds 3.
To find f(2), we substitute x = 2 into the function: f(2) = 2(2) + 3 = 7.
Similarly, to find f(-1), we substitute x = -1: f(-1) = 2(-1) + 3 = 1.

Determining if a Relation is a Function: The Vertical Line Test Explained

The vertical line test is a simple but powerful visual tool for determining whether a graph represents a function. As mentioned earlier, a graph represents a function if and only if no vertical line intersects the graph more than once.

How to apply the Vertical Line Test:

Visualize or Draw Vertical Lines: Imagine drawing vertical lines across the entire graph.
Check for Intersections: For each vertical line, count the number of times it intersects the graph.
Interpret the Results:
- If every vertical line intersects the graph at most once (i.e., zero or one time), then the graph represents a function.
- If any vertical line intersects the graph more than once (i.e., two or more times), then the graph does not represent a function.

Why does the Vertical Line Test work?

The vertical line test directly relates to the definition of a function. A vertical line represents a specific x-value. If the vertical line intersects the graph at more than one point, it means that that single x-value is associated with multiple y-values, violating the requirement that each input (x) must have a unique output (y).

Examples:

Parabola (y = x^2): A parabola opens upwards or downwards. Any vertical line will intersect the parabola at most once. Therefore, a parabola represents a function.
Line (y = mx + b): A straight line (except for a vertical line itself) will always pass the vertical line test. It represents a function.
Circle (x^2 + y^2 = r^2): A circle will fail the vertical line test. For example, a vertical line drawn through the center of the circle will intersect the circle at two points (one above and one below the center). Therefore, a circle does not represent a function.
Vertical Line (x = a): A vertical line fails the vertical line test trivially. Any vertical line drawn on top of it will intersect it infinitely many times. Moreover, every point on the line has the same x-value, but different y-values. Therefore, a vertical line does not represent a function.

Domain and Range of Relations and Functions

Understanding the domain and range of a relation or function is crucial for fully describing its behavior.

Domain: The domain is the set of all possible input values (x-values) for which the relation or function is defined.
Range: The range is the set of all possible output values (y-values) that the relation or function can produce.

Finding the Domain:

Set of Ordered Pairs: The domain is simply the set of all first elements (x-values) in the ordered pairs.
Graph: The domain is the set of all x-values that have a corresponding y-value on the graph. Visually, project the graph onto the x-axis; the domain is the interval or set of intervals covered by the projection.
Equation: This is often the trickiest. Consider any restrictions on the x-values:
- Division by Zero: The denominator of a fraction cannot be zero. Therefore, any x-value that would make the denominator zero must be excluded from the domain.
- Square Roots of Negative Numbers: In the real number system, you cannot take the square root (or any even root) of a negative number. Therefore, any x-value that would make the expression under the square root negative must be excluded from the domain.
- Logarithms of Non-Positive Numbers: You cannot take the logarithm of a non-positive number (zero or negative). Therefore, any x-value that would make the argument of a logarithm non-positive must be excluded from the domain.

Finding the Range:

Set of Ordered Pairs: The range is simply the set of all second elements (y-values) in the ordered pairs.
Graph: The range is the set of all y-values that have a corresponding x-value on the graph. Visually, project the graph onto the y-axis; the range is the interval or set of intervals covered by the projection.
Equation: Finding the range from an equation can be more challenging than finding the domain. Sometimes, you can solve the equation for x in terms of y and then consider restrictions on y, similar to how you find the domain. However, for many functions, it's best to analyze the function's behavior and consider its graph to determine the range.

Examples:

f(x) = 2x + 1:
- Domain: All real numbers (represented as (-∞, ∞) or ℝ) because there are no restrictions on x.
- Range: All real numbers ((-∞, ∞) or ℝ) because for any y-value, you can find an x-value that produces it.
g(x) = 1/(x - 3):
- Domain: All real numbers except x = 3 (represented as (-∞, 3) ∪ (3, ∞)) because division by zero is undefined when x = 3.
- Range: All real numbers except y = 0 (represented as (-∞, 0) ∪ (0, ∞)). The function can get arbitrarily close to 0, but it will never actually equal 0.
h(x) = √(x + 2):
- Domain: x ≥ -2 (represented as [-2, ∞)) because the expression under the square root must be non-negative.
- Range: y ≥ 0 (represented as [0, ∞)) because the square root function always returns a non-negative value.
k(x) = x^2:
- Domain: All real numbers ((-∞, ∞) or ℝ).
- Range: y ≥ 0 (represented as [0, ∞)) because the square of any real number is non-negative.

Practice Questions and Answer Key

Here are some practice questions to test your understanding of relations and functions. Answers are provided below.

Question 1:

Which of the following relations is a function?

a) {(1, 2), (2, 3), (1, 4), (3, 5)} b) {(4, 6), (5, 7), (6, 8), (7, 9)} c) {(0, 0), (1, 1), (2, 2), (0, 3)} d) {(a, b), (c, d), (a, e), (f, g)}

Question 2:

Does the equation x^2 + y = 4 represent y as a function of x?

Question 3:

What is the domain of the function f(x) = √(x - 5)?

Question 4:

What is the range of the function g(x) = x^2 + 3?

Question 5:

A graph shows a curve that passes through the points (1, 2), (2, 4), (3, 6), and (1, 8). Does this graph represent a function? Explain using the vertical line test.

Question 6:

Determine the domain and range of the relation represented by the set of ordered pairs: {(-2, 1), (-1, 0), (0, 1), (1, 2), (2, 3)}

Question 7:

Is the relation x = |y| a function? Explain why or why not.

Question 8:

Given the function f(x) = (x + 1) / (x - 2), find f(0), f(-1), and f(3).

Question 9:

Sketch the graph of y = |x|. Does it represent a function? What is its domain and range?

Question 10:

A taxi company charges $2.50 as a base fare and $0.50 per mile. Write a function that represents the total cost C(m) as a function of the number of miles m. What is the cost for a 10-mile ride?

Answer Key:

b) {(4, 6), (5, 7), (6, 8), (7, 9)} (Each x-value has a unique y-value.)
Yes. Solving for y, we get y = 4 - x^2. For each x-value, there is only one corresponding y-value.
x ≥ 5 (or [5, ∞))
y ≥ 3 (or [3, ∞))
No. The vertical line x = 1 intersects the graph at two points, (1, 2) and (1, 8), violating the vertical line test.
Domain: {-2, -1, 0, 1, 2}. Range: {0, 1, 2, 3}
No. For example, if x = 2, then y could be 2 or -2. Therefore, one x-value has two corresponding y-values.
f(0) = -1/2, f(-1) = 0, f(3) = 4
The graph of y = |x| is a V-shape, with the vertex at the origin (0, 0). It represents a function. Domain: All real numbers. Range: y ≥ 0.
C(m) = 2.50 + 0.50m. For a 10-mile ride: C(10) = 2.50 + 0.50(10) = $7.50

Advanced Topics: Types of Functions

While the basics are essential, understanding different types of functions can further enhance your knowledge. Here are a few key categories:

Linear Functions: Functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Their graphs are straight lines.
Quadratic Functions: Functions of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Their graphs are parabolas.
Polynomial Functions: Functions that involve only non-negative integer powers of x. Examples include x^3 + 2x^2 - x + 5.
Rational Functions: Functions that are ratios of two polynomials. Examples include (x + 1) / (x^2 - 4).
Exponential Functions: Functions of the form f(x) = a^x, where a is a constant and a > 0.
Logarithmic Functions: Functions that are the inverse of exponential functions. They are written as f(x) = log_a(x).
Trigonometric Functions: Functions like sine (sin), cosine (cos), and tangent (tan) that relate angles of a triangle to the ratios of its sides.
Absolute Value Functions: Functions of the form f(x) = |x|, which return the non-negative value of x.
Piecewise Functions: Functions defined by different formulas for different intervals of the domain.

Each type of function has its own unique properties, graph, domain, and range. Understanding these characteristics is crucial for analyzing and solving more complex problems.

Tips for Test Preparation

Review Definitions: Make sure you have a solid understanding of the definitions of relation, function, domain, and range.
Practice, Practice, Practice: Work through as many practice problems as possible. The more you practice, the more comfortable you'll become with the concepts.
Understand the Vertical Line Test: Be able to apply the vertical line test to determine if a graph represents a function.
Identify Domain Restrictions: Practice finding the domains of functions, paying close attention to division by zero, square roots of negative numbers, and logarithms of non-positive numbers.
Know Key Function Types: Familiarize yourself with the characteristics of linear, quadratic, and other common types of functions.
Work Through Examples: Study worked-out examples to see how the concepts are applied in different situations.
Review Your Notes: Go over your class notes and textbook examples to reinforce your understanding.
Seek Help When Needed: Don't hesitate to ask your teacher or classmates for help if you're struggling with a particular concept.

By mastering the concepts outlined in this study guide and practicing diligently, you'll be well-prepared for your Unit 3 test on relations and functions and build a strong foundation for future mathematical studies. Good luck!