Lesson 1.4

Domain of Basic Algebraic Functions

In the real number system, not every input is allowed. Learn the two "Commandments" that forbid certain mathematical operations.

1

Introduction

Prerequisite Connection: To find a domain, you need to be proficient in solving linear inequalities (e.g., ), a skill from Algebra 1.

Today's Increment: We define the Domain as the set of all "legal" inputs. By default, the domain is All Real Numbers (), unless one of two specific "Red Flags" appears: division by zero or an even root of a negative.

Why This Matters for Calculus: You cannot differentiate or integrate a function where it doesn't exist. Identifying domain breaks is the first step in finding vertical asymptotes and discontinuities.

2

Explanation of Key Concepts

The Two "Red Flags"

Red Flag #1: Division

You cannot divide by zero. It is undefined.

The Rule
Red Flag #2: Even Roots

You cannot take the square root (or 4th, 6th root) of a negative number in the Real system.

The Rule

The "Green Light" Functions

If a function has NO division and NO even roots, its domain is automatically:

This applies to:

  • Polynomials (e.g., )
  • Odd Roots (e.g., , because you can cube root a negative!)
  • Exponential functions (e.g., )
3

Worked Examples

Level: Basic

Example 1: Rational Function

Find the domain of .

Step 1: Identify Red Flag
Division detected. The denominator cannot be zero.
Step 2: Set Restriction
Step 3: Solve
Interval Notation:
Level: Intermediate

Example 2: Square Root Function

Find the domain of .

Step 1: Identify Red Flag
Even root detected. The "inside" must be non-negative.
Step 2: Set Restriction
Step 3: Solve
Interval Notation:
Level: Advanced (Calculus Prep)

Example 3: Combined Restrictions

Find the domain of .

Red Flag Conflict

1. Denominator rule:
2. Root rule:

Combined Rule: Inside (Strictly Positive)

Step 1: Set Combined Inequality
Step 2: Solve carefully
Warning: Dividing by negative!
Interval Notation:
4

Common Pitfalls

  • Restricting Odd Roots:

    The cube root of -8 is -2. It works fine. Only even roots (square, 4th) restrict the domain.

  • Forgetting to Flip the Sign:

    In inequalities like , when you divide by -3, the alligator flips: .

5

Real-World Application

Physics: Physical Constraints

In math, "domain" is abstract. In physics, domain is reality.

  • Time constraint: (We cannot go back in time).
  • Mass constraint: (Objects cannot have negative mass).
  • Dimension constraint: Lengths, widths, and radii must be positive.

Engineers must always define the "Physical Domain" of their functions to prevent computers from simulating impossible scenarios.

6

Practice Quiz

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