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.

Introduction

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

Past Knowledge

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

Today's Goal

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.

Future Success

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.

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., )

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:

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: .

Real-Life Applications

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.

Practice Quiz

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