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The Domain of an Algebraic Expression

The Domain of an Algebraic Expression. In general, an algebraic expression may not be defined for all values of the variable. The domain of an algebraic expression is the set of real numbers that the variable is permitted to have.

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The Domain of an Algebraic Expression

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  1. The Domain of an Algebraic Expression • In general, an algebraic expression may not be defined for all values of the variable. • The domain of an algebraic expression is the set of real numbers that the variable is permitted to have. • The table in the margin givessome basic expressions andtheir domains.

  2. Example 1 – Finding the Domain of an Expression • Find the domains of the following expressions. • (a) 2x2 + 3x – 1 • (b) • (c)

  3. Example 1 – Solution • (a) The domain is the set of all real numbers. • (b) We first factor the denominator. • the denominator is zero when x = 2 or 3, the expression is not defined there. The domain is {x | x  2 and x 3}.

  4. Example 1 – Solution • cont’d • (c) For the numerator to be defined, we must have x  0. Also, we cannot divide by zero, so x 5. • Thus, the domain is {x | x  0 and x 5}.

  5. Example 2 – Simplifying Rational Expressions by Cancellation • Simplify: • Solution: • Factor • Cancel common factors

  6. Example 3 – Multiplying Rational Expressions • To multiply rational expressions, we multiply their numerators and multiply their denominators. • Solution: • We first factor. • Factor • Cancel common • factors

  7. Example 4 – Dividing Rational Expressions • To Divide: invert 2nd term then multiply • Solution: • Invert and multiply • Factor • Cancel common • factors

  8. Example 5 – Adding and Subtracting Rationals: Common Denominator!!! • (a) (b) • Solution: • (a) Here the LCD is simply the product (x – 1)(x + 2). • Write fractionsusing LCD • Add fractions • Combine termsin numerator

  9. Example 5 – Solution • cont’d • (b) The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2 is (x – 1)(x + 1)2. • Factor • Combine fractions using LCD • Distributive Property • Combine terms in numerator

  10. Example 6 – Simplifying a Compound Fraction • Simplify: • Solution 1:We combine the terms in the numerator into a single fraction. We do the same in the denominator. • Then we invert and multiply.

  11. Example 6 – Solution • cont’d • Soln 2: We find the LCD of all the little fractions, then multiply every term by it. Here the LCD is xy. • Multiply numerator • and denominator by xy

  12. Example 6 – Solution • cont’d • Simplify • Factor

  13. Rationalizing the Denominator or the Numerator • If a fraction has a denominator of the form A + B , we may rationalize the denominator by multiplying numerator and denominator by the conjugate radical A – B . • This works because, • (A + B )(A – B ) = A2 – B2C

  14. Example 9 – Rationalizing the Denominator • Rationalize the denominator: • Solution:We multiply both the numerator and the denominator by the conjugate radical of 1 + , which is 1 –. • Multiply numerator and • denominator by the • conjugate radical • Special Product Formula 1

  15. Example 9 – Solution • cont’d

  16. Avoiding Common Errors • The following table states several properties of multiplication and illustrates the error in applying them to addition.

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