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Mapping and Functions: Understanding the Concepts Through Examples

Learn about mappings, functions, isometry, and one-to-one functions through detailed examples and explanations. Practice problems included.

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Mapping and Functions: Understanding the Concepts Through Examples

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  1. 14-1 Mappings and Functions Answer to Warm Up Similar = Proportional sides Congruent = Equal sides

  2. Mapping: A correspondence between a set of points (x,y) → (x – 2, y + 1) Pre-image Image (0,0) → (-2,1) (2,3) → (0,4) (3,1) → (1,2) T T’ Isometry: Pre-image and image are congruent T’ T

  3. TOO (use previous pre-image) • Using the mapping (x,y) → (2x, -y) • Find and graph the image. • Is it an isometry? Answers: • (0,0) (4,-3) (6,-1) • No

  4. Example • Given (x,y) → (2x, -y) • Find the image of (-2,-3) (2•-2,-(-3)) (-4,3) • Find the pre-image of (6,-1) (2x = 6, -y = -1) (3,1)

  5. TOO: (x,y) → (x + 1,y + 1) • Find the image of: • (3,4) • (5,1) • (-1,-3) • Find the pre-image of: • (0,0) • (4,5) • Answers • (4,5) • (6,2) • (0,-2) • (-1,-1) • (3,4)

  6. Function: A correspondence between a set of numbers f: x → 2x – 5 Pre-image Image Find the image of 0. 2(0) – 5 0 – 5 -5 Find the pre-image of -1. 2x – 5 = -1 2x = 4 x = 2

  7. Alternate Example f(x) = 2x2 f(1) = 2(1)2 = 2(1) = 2 f(-1) = 2(-1)2 = 2(1) = 2 • One-to-One function: each image matches to exactly one pre-image • Not one-to-one, image of “2” maps to two pre-images (1 and -1).

  8. TOO: f:x → 7 – 2x • Find the image of: • 5 • -2 • Find the pre-image of: • 13 • -1 • Is it one-to-one? • Answers: • -3 • 11 • -3 • 4 • Yes

  9. Homework • Page 574 #1-10

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