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Understanding Functions in Mathematics

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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Understanding 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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