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Functions in Mathematics

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Explore various ways to represent functions, analyze domains and ranges, and practice composition of functions in math. Learn function definitions and relations with examples in this comprehensive guide.

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Functions in Mathematics

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  1. Bell work: • Which of these are ways to represent a function? • f(x)=x+3 • f(x)=+5 • (0,1),(1,2),(2,3),(3,4),(4,5)… • Double a number and add six • The image to the right

  2. What is a function? • In your groups, arrive at a shared definition of what a ‘function’ is. When you’re finished, turn in your group’s definition(with all member’s names) • We will discuss different definitions as a class.

  3. 1.1: Functions and Relations

  4. Function, relation, or neither?

  5. Function, relation, or neither?

  6. Relations vs functions • A relation is some relationship between inputs and outputs • A function is a relation that matches each input WITH ONLY ONE output

  7. Is this a function?

  8. Which are not functions?

  9. Domain and Range • Domain is the set of all inputs(aka x-values) • Range is the set of all outputs (aka y-values)

  10. What is the domain and range of these functions? • f(x) = 5x+7 • g(x) = 5 • h(x) = • r(x) =

  11. 1.2: Composition of Functions

  12. What is a composition of functions?

  13. Adding functions • If f(x)=+5, and g(x) = x + 7, what is f(4) + g(4) ? • If h(x)=+12, what is h(4) ?

  14. Adding functions • If f(x)=-2, and g(x) = , what is f(2) + g(2) ? • If h(x)=-2, what is h(2) ?

  15. Subtracting functions • If f(x)=+5, and g(x) = x + 7, what is f(4) - g(4) ? • If f(x)=-2, and g(x) = , what is f(2) - g(2) ?

  16. Multiplying functions • If f(x)=+5, and g(x) = x + 7, what is f(4) * g(4) ? • If f(x)=-2, and g(x) = , what is f(2) * g(2) ?

  17. Dividing functions • If f(x)=+5, and g(x) = x + 7, what is f(4) / g(4) ? • If f(x)=-2, and g(x) = , what is f(2) / g(2) ?

  18. IN GENERAL…what can we say about… • f(x)+g(x)=… • f(x)-g(x)=… • f(x)*g(x)=… • f(x)/g(x)=… (f+g)(x) (f-g)(x) (f*g)(x) (f/g)(x) As long as g(x) does not equal zero

  19. Evaluating functions • If f(x) = 12x+5, what is f(3)? • If f(x) = 12x+5, what is f(y)? • If f(x) = 12x+5, what is f( )?

  20. Evaluating functions • If f(x) = 12x+5, what is f(3+7)?

  21. Composition of functions • The previous example was a composition of functions. Basically, everywhere we had a variable, we replace it with an entirely new function. • This is denoted as ᴑ(x), or as

  22. Find ᴑ(x) for

  23. Homework: • Section 1.1, pages 9-11: 2, 10, 11, 13-15, 38-40, 42, 44, 47 • Section 1.2, pages 17-18: 5-8, 14, 18, 19, 22-24

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