Unit 3 Test Study Guide Relations And Functions Answer Key

Unit 3 Test Study Guide: Relations and Functions - Answer Key

This comprehensive study guide covers key concepts related to relations and functions, providing answers and explanations to help you ace your Unit 3 test. We'll delve into the core definitions, explore different representations, and tackle various problem types, ensuring you're fully prepared. Remember to consult your class notes and textbook for additional examples and practice problems.

What are Relations and Functions?

Understanding the fundamental difference between relations and functions is crucial.

Relation: A relation is simply a set of ordered pairs (x, y). Think of it as a connection or association between two sets of values. There are no restrictions on how many times an x-value can appear, or which y-values are associated with it.

Function: A function is a special type of relation where each x-value (input) is associated with only one y-value (output). This "one-to-one" or "many-to-one" mapping is the defining characteristic of a function.

Identifying Functions: The Vertical Line Test

The easiest way to visually determine if a relation is a function is the Vertical Line Test (VLT).

• How it works: If you can draw a vertical line anywhere on the graph of a relation and it intersects the graph at more than one point, then the relation is not a function. If every vertical line intersects the graph at most once, then it is a function.

Example: A parabola (like y = x²) passes the VLT and is a function. However, a circle (like x² + y² = 1) fails the VLT because vertical lines can intersect it at two points, making it a relation but not a function.

Representing Relations and Functions

Relations and functions can be represented in several ways:

• Ordered Pairs: {(1, 2), (2, 4), (3, 6)}
• Tables:

• Graphs: A visual representation on the coordinate plane.
• Mappings: Uses arrows to show the relationship between x and y values.
• Equations: y = 2x (this is a function because for every x, there's only one y)

Types of Functions

Various types of functions possess unique characteristics and behaviors:

• Linear Functions: Represented by the equation y = mx + b, where 'm' is the slope and 'b' is the y-intercept. They create straight lines on a graph. Key features to understand include slope, intercepts, and how to find the equation from points or a graph.

• Quadratic Functions: Represented by the equation y = ax² + bx + c, where 'a', 'b', and 'c' are constants. They create parabolas. Key concepts here are vertex, axis of symmetry, x-intercepts (roots), y-intercept, and how to find the equation in various forms (standard, vertex, factored).

• Polynomial Functions: These are functions with multiple terms, each involving a variable raised to a non-negative integer power. Quadratic functions are a subset of polynomial functions. Understanding the degree of the polynomial (highest power of x) is crucial for predicting its behavior.

• Exponential Functions: These functions have a variable in the exponent, like y = a * bˣ. They exhibit rapid growth or decay. Understanding the base 'b' and its effect on growth/decay is key.

• Rational Functions: Functions of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials. They often have asymptotes (vertical and horizontal lines the graph approaches but doesn't touch).

Domain and Range

• Domain: The set of all possible x-values (inputs) for a relation or function. Sometimes restricted by the context of the problem (e.g., you can't have negative lengths).

• Range: The set of all possible y-values (outputs) for a relation or function.

Finding Domain and Range

Methods for finding the domain and range depend on the representation of the function:

• From a graph: Visually examine the x-values and y-values covered by the graph.
• From a set of ordered pairs: List the unique x-values for the domain and the unique y-values for the range.
• From an equation: Consider any restrictions on the input (x) values. For example:
  • Avoid dividing by zero: The denominator of a rational function cannot be zero.
  • Avoid even roots of negative numbers: The expression inside a square root must be non-negative.

Function Notation

Function notation, typically f(x), represents the output of a function f when the input is x. This allows for easy evaluation and manipulation of functions. For instance, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.

Function Operations

Functions can be combined using various operations:

• Addition: (f + g)(x) = f(x) + g(x)
• Subtraction: (f - g)(x) = f(x) - g(x)
• Multiplication: (f * g)(x) = f(x) * g(x)
• Division: (f / g)(x) = f(x) / g(x) (provided g(x) ≠ 0)
• Composition: (f ∘ g)(x) = f(g(x)) – This means substituting the entire function g(x) into the function f(x).

Inverse Functions

An inverse function, denoted f⁻¹(x), "undoes" the original function f(x). If f(a) = b, then f⁻¹(b) = a. Not all functions have inverses; only one-to-one functions (those that pass both the vertical and horizontal line tests) possess inverses. Finding the inverse involves swapping x and y in the equation and solving for y.

Sample Problems and Solutions

Let's work through some example problems to solidify your understanding:

Problem 1: Is the relation {(1, 2), (2, 4), (3, 6), (1, 3)} a function? Why or why not?

Answer: No. The x-value 1 is associated with two different y-values (2 and 3), violating the definition of a function.

Problem 2: Find the domain and range of the function f(x) = √(x - 4).

Answer:

• Domain: The expression inside the square root must be non-negative: x - 4 ≥ 0, so x ≥ 4. The domain is [4, ∞).
• Range: Since the square root of a non-negative number is always non-negative, the range is [0, ∞).

Problem 3: Given f(x) = x² and g(x) = x + 1, find (f ∘ g)(x).

Answer: (f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)² = x² + 2x + 1

Problem 4: Find the inverse of the function f(x) = 3x - 6.

Answer:

1. Replace f(x) with y: y = 3x - 6
2. Swap x and y: x = 3y - 6
3. Solve for y: x + 6 = 3y => y = (x + 6)/3
4. Replace y with f⁻¹(x): f⁻¹(x) = (x + 6)/3

Problem 5: Determine whether the following graph represents a function. (Imagine a graph here – for testing purposes, consider a simple parabola or a circle. A parabola is a function, a circle is not.)

Answer: (The answer depends on the graph presented. Apply the Vertical Line Test.)

Problem 6: Identify the type of function represented by y = 2ˣ.

Answer: Exponential function.

Problem 7: What is the slope of the linear function y = -2x + 5?

Answer: The slope is -2.

Further Practice and Resources

This study guide provides a solid foundation. To reinforce your understanding, work through additional practice problems from your textbook or online resources. Focus on practicing different types of problems, including those involving different representations of functions and operations on functions. Remember to review your class notes and seek clarification from your teacher if needed. Good luck with your Unit 3 test!