Unlocking the Domain: Finding Function Domains Algebraically and Graphically

PreAlgebra Grades High School 4:17 Video
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Lesson Description

Master the concept of domain for functions, learning to identify and express it through equations, inequalities, interval notation, and graphical analysis. This lesson uses real examples to solidify your understanding.

Video Resource

How to Find the Domain of a Function Given the Equation or Graph

Mario's Math Tutoring

Duration: 4:17
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Key Concepts

  • Domain of a Function
  • Interval Notation
  • Inequality Notation
  • Restrictions on Domain (Division by Zero, Square Root of Negatives)
  • Graphical Analysis of Domain

Learning Objectives

  • Identify restrictions on the domain of a function based on its equation.
  • Express the domain of a function using interval and inequality notation.
  • Determine the domain of a function from its graph.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definition of a function and introduce the concept of domain as the set of all possible input values (x-values) for which the function is defined. Preview the types of functions where domain restrictions are common (rational functions, functions with radicals).
  • Domain Restrictions: Rational Functions (10 mins)
    Explain that division by zero is undefined. Therefore, in rational functions (functions with a variable in the denominator), any x-value that makes the denominator zero must be excluded from the domain. Work through Example 1 from the video (y = 3/(2x-1)), demonstrating how to find the restricted value and express the domain using both inequality notation (x ≠ 1/2) and interval notation ((-∞, 1/2) ∪ (1/2, ∞)).
  • Domain Restrictions: Radical Functions (15 mins)
    Explain that the square root (or any even root) of a negative number is not a real number. Therefore, in functions with even roots, the expression under the radical must be greater than or equal to zero. Work through Example 2 from the video (y = √(5x-3)), demonstrating how to set up and solve the inequality (5x-3 ≥ 0), and express the domain using both inequality notation (x ≥ 3/5) and interval notation ([3/5, ∞)). Emphasize that odd roots (like cube roots) do not have this restriction.
  • Domain from a Graph (10 mins)
    Explain that the domain of a function from its graph is the set of all x-values that have a corresponding y-value on the graph. Demonstrate using Example 3 from the video. Show how to visually scan the graph from left to right (negative to positive x-values) to identify the interval(s) where the graph exists. Emphasize the use of closed vs. open intervals based on whether endpoints are included or excluded.
  • Practice and Review (10 mins)
    Provide additional practice problems covering both algebraic and graphical determination of domain. Encourage students to work independently or in pairs, and review the solutions as a class.

Interactive Exercises

  • Equation Domain Challenge
    Students are given a series of equations (rational functions and functions with radicals) and must determine the domain using algebraic methods. They should express their answers in both inequality and interval notation.
  • Graph Domain Discovery
    Students are presented with various graphs of functions and must identify the domain of each function by visually analyzing the graph. They should express their answers in interval notation.

Discussion Questions

  • Why is it important to understand the domain of a function?
  • Can a function have no domain restrictions? If so, give an example.
  • How does the concept of domain relate to the graph of a function?
  • Explain in your own words how interval notation and inequality notation represent the same domain. What are the benefits of each?

Skills Developed

  • Algebraic manipulation
  • Inequality solving
  • Interval notation
  • Graphical analysis
  • Critical Thinking

Multiple Choice Questions

Question 1:

Which of the following is NOT a reason for a function to have a restricted domain?

Correct Answer: Cube root of a negative number

Question 2:

The domain of the function f(x) = 1/(x+3) is:

Correct Answer: (-∞, -3) ∪ (-3, ∞)

Question 3:

The domain of the function g(x) = √(2x - 6) is:

Correct Answer: [3, ∞)

Question 4:

Which interval notation represents all real numbers greater than or equal to 5?

Correct Answer: [5, ∞)

Question 5:

Which inequality represents the domain (-∞, 2)?

Correct Answer: x < 2

Question 6:

The graph of a function has a break at x = 4. Which of the following could be a potential domain?

Correct Answer: (-∞, 4) ∪ (4, ∞)

Question 7:

What is the domain of a function represented graphically by a line segment starting at (1,2) and ending at (5,6), inclusive?

Correct Answer: [1,5]

Question 8:

Which of the following best describes how to find domain from a graph?

Correct Answer: Read from left to right

Question 9:

What is the domain of f(x) = √(-x)?

Correct Answer: x ≤ 0

Question 10:

The domain of a function is all real numbers except x = -2 and x = 5. Which interval notation represents this domain?

Correct Answer: (-∞, -2) ∪ (-2, 5) ∪ (5, ∞)

Fill in the Blank Questions

Question 1:

The set of all possible input values for a function is called the __________.

Correct Answer: domain

Question 2:

In interval notation, a curved bracket or parenthesis indicates that the endpoint is __________.

Correct Answer: not included

Question 3:

When dealing with square roots, the expression under the radical must be __________ or equal to zero to avoid imaginary numbers.

Correct Answer: greater than

Question 4:

The domain of the function f(x) = √x is [__________, ∞).

Correct Answer: 0

Question 5:

To find the domain of a rational function, set the __________ equal to zero and solve for x to find the excluded values.

Correct Answer: denominator

Question 6:

The function f(x) = |x| has a domain of __________.

Correct Answer: all real numbers

Question 7:

When scanning a graph to determine the domain, we visually move from __________ to __________.

Correct Answer: left, right

Question 8:

A function with an _____ root does not have the restriction of not having negative values

Correct Answer: odd

Question 9:

The inequality x > -3 represents the interval __________.

Correct Answer: (-3, ∞)

Question 10:

For even-rooted radicals, the radicand must be _____ or equal to zero

Correct Answer: greater than

Educational Standards

PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.3:Interpret characteristics of graphs and tables for a function that models a relationship between two quantities in terms of the quantities. PC.F.1.4:Describe end behavior, asymptotic behavior, and points of discontinuity.

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