Overview
This lecture explains how to perform basic algebraic operations—addition, subtraction, multiplication, and division—on two linear functions: F(x) = 5x + 3 and G(x) = 2x - 1.
Given Functions
- F(x) = 5x + 3
- G(x) = 2x - 1
Addition of Functions
- (F + G)(x) = F(x) + G(x) = (5x + 3) + (2x - 1)
- Combine like terms: 5x + 2x = 7x; 3 - 1 = 2
- Final result: (F + G)(x) = 7x + 2
Subtraction of Functions
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(F - G)(x) = F(x) - G(x) = (5x + 3) - (2x - 1)
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Distribute the minus sign: (5x + 3) - 2x + 1
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Combine like terms: 5x - 2x = 3x; 3 + 1 = 4
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Final result: (F - G)(x) = 3x + 4
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(G - F)(x) = G(x) - F(x) = (2x - 1) - (5x + 3)
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Distribute the minus sign: 2x - 1 - 5x - 3
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Combine like terms: 2x - 5x = -3x; -1 - 3 = -4
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Final result: (G - F)(x) = -3x - 4
Multiplication of Functions
- (F * G)(x) = (5x + 3)(2x - 1)
- Multiply terms: (5x)(2x) = 10x²; (5x)(-1) = -5x; (3)(2x) = 6x; (3)(-1) = -3
- Combine like terms: -5x + 6x = x
- Final result: (F * G)(x) = 10x² + x - 3
Division of Functions
- (F / G)(x) = (5x + 3)/(2x - 1)
- (G / F)(x) = (2x - 1)/(5x + 3)
Key Terms & Definitions
- Function — a relation that assigns each input exactly one output.
- Like terms — terms in algebraic expressions that have the same variable part.
- Polynomial multiplication — distributing each term in the first polynomial by each term in the second.
Action Items / Next Steps
- Practice these operations with different functions to reinforce understanding.