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Understanding Real Numbers, Relations, and Functions

Explore subsets of real numbers, relations on the Cartesian plane, and functions through examples and exercises. Learn about natural, whole, integers, rational, and irrational numbers. Practice finding domains, ranges, and function values.

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Understanding Real Numbers, Relations, and Functions

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  1. Chapter 1: Number Patterns1.1: Real Numbers, Relations, and Functions Essential Question: What are the subsets of the real numbers? Give an example of each.

  2. 1.1: Real Numbers, Relations, and Functions • Real Numbers • Natural Numbers: • Whole Numbers: • Integers: • Rational Numbers: Can be expressed as a ratio • Irrational Numbers: No way to simplify the number • Non-terminating, non-repeating decimals 1, 2, 3, 4 … 0, 1, 2, 3, 4 … … -3, -2, -1, 0, 1, 2, 3, …

  3. 1.1: Real Numbers, Relations, and Functions • All real numbers are either rational or irrational • Rational Numbers • Integers • Whole Numbers • Natural Numbers • Irrational Numbers

  4. 1.1: Real Numbers, Relations, and Functions • Cartesian plane: another name for the coordinate plane • Numbers are placed on the coordinate plane using ordered pairs • Ordered pairs are in the form (x, y) • Scatter plot → Data placed on a coordinate plane • Domain of a relation → possible x values • Range of a relation → possible y values

  5. 1.1: Real Numbers, Relations, and Functions • Example 2: Domain and Range of a Relation Find the relation’s domain and range • Answer: We can use the ordered pair (height, shoe size) for our relation. This give us 12 ordered pairs: (67,8.5),(72,10),(69,12),(76,12),(67,10),(72,11), (67,7.5),(62.5,5.5),(64.5,8),(64,8.5),(62,6.5),(62,6) • Domain: {62, 62.5, 64, 64.5, 67, 69, 72, 76} • Range: {5.5, 6, 6.5, 7.5, 8, 8.5, 10, 11, 12}

  6. 1.1: Real Numbers, Relations, and Functions • Functions → a method where the 1st coordinate of an ordered pair represents an input, and the 2nd represents an output • Each input corresponds to one AND ONLY ONE output • Example 4: Identifying a Function Represented Numerically • {(0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3)} • {(0,0),(1,1),(-1,-1),(4,2),(-4,2),(9,3),(-9,3)} • {(0,0),(1,1),(-1,-1),(4,2),(-4,-2),(9,3),(-9,-3)}

  7. 1.1: Real Numbers, Relations, and Functions • Example 5: Finding Function Values from a Graph / Figure 1.1-8 • On board • Functional Notation • f(x) denotes the output of the function f produced by the input x • y= f(x) read as “y equals f of x”

  8. 1.1: Real Numbers, Relations, and Functions • Functional Notation • f = name of function • x = input number • f(x) = output number = • = directions on what to do with the input

  9. 1.1: Real Numbers, Relations, and Functions • Functional Notation (Example 6) • For h(x) = x2 + x – 2, find each of the following • h(-2) = (-2)2 + (-2) – 2 = 4 – 2 – 2 = 0 • h(-a) = (-a)2 + (-a) – 2 = a2 – a – 2

  10. 1.1: Real Numbers, Relations, and Functions • Assignment • Page 10-12 • 1-33, odd problems

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