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Schiopu Alina-Elena clasa a 11 a A 27 septembrie 2011

Schiopu Alina-Elena clasa a 11 a A 27 septembrie 2011. Functions-generalization. Math in real life!!!. Math is everywhere in real life .

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Schiopu Alina-Elena clasa a 11 a A 27 septembrie 2011

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  1. Schiopu Alina-Elenaclasa a 11 a A27 septembrie 2011 Functions-generalization

  2. Math in real life!!! • Mathiseverywhere in real life. • Functions are commonly used in math,science,andengineering.Forexample,if a bowman wants to shoot an arrow to the target 80 meters away,and the question asks you about the power the bowman should use at different location.

  3. Vocabulary • Input=value • Output=image • Relationship=relate,connection • Input,output,relationship} Function • f(x)=f of x • x²=x squared • ‘=‘=equals • Set(multime)=a collection of things,a group of elements. • Domain(domeniu de definitie) • Codomain(codomeniu) • Range=Imf • Grapher(graficulfunctiei)

  4. Increasing functions(functii crescatoare) • Decreasing functions(functii descrecatoare) • Constant functions • Maximum,minimum(of a function) • Even functions(fct.pare) • Odd functions(fct.impare) • Composite functions(fct.compuse) • Injective,surjective(also called’onto’),bijective • Inverse function(fct.inversa) • f(x₁)≥f(x₂) = f of x₁ is larger than(or equal to)f(x₂) • f(x₁)≤f(x₂)=f of x₁ is smaller than(or equal to)f(x₂) • Monotonic functions • Diagram.

  5. Theoretical notionsDefinitions and important results • What is a function? • A function relates each element of a set with exactly one element of another set(possibly the same set). • Examples: x² is a function; x³+1 is also a function;sine,cosineand tangent are functions used in trigonometry etc.

  6. Domain,codomain and range • The set x is called the Domain. • The set y is called the Codomain. • The Range is the set of elements that get pointed to in y. RELATIONSHIP INPUT OUTPUT f,g f(x),g(t) x,t Range elements Domain elements

  7. x The function f(x)=x²represented by Venn Diagrams y 1 3 2 1 4 9 A function is a special type of relation where every element in the domain is included ,and any input produces only one output. ‘each element’ means that every element in x is related to some element iny.

  8. Increasing and decreasing functions • A function is increasingif the y-value increases as the x-value increases. When a function is increasing? For a function y=f(x): when x₁<x₂ then f(x₁)≤f(x₂) INCREASING when x₁<x₂then f(x₁)<f(x₂)STRICTLY INCREASING There has to be true for any x₁,x₂,not just some nice ones you choose. A function is decreasing when the y-value decreases as the x-value increases . When a function is decreasing? For a function y=f(x): when x₁<x₂ then f(x₁)≥f(x₂) DECREASING when x₁<x₂ then f(x₁)>f(x₂) STRICTLYDECREASING.

  9. Even functions A function is even when: f(-x)=f(x)for all x. A function is even if there is symmetry about the y-axis!!!! Odd functions A function is odd when: f(-x)=-f(x) for all x. Cosine and sine functions f(x)=cos(x) is an even function. f(x)=sin(x) is an odd function. A function is odd if there is symmetry about the Origin.

  10. Composition of function ‘Function Composition’is applying one function to the results of another: f() Theresult of f() issentthrough g() Itiswritten:(g˚f)(x) Which means:g(f(x)). Example:f(x)=2x+3 and g(x)=x² (g°f)(x)=(2x+3)². g()

  11. Injective,surjective,bijective • Three important kinds of function are the injections (or one-to-one functions), which have the property that if ƒ(a) = ƒ(b) then a must equal b; the surjections (or onto functions), which have the property that for every y in the codomain there is an x in the domain such that ƒ(x) = y; and the bijections, which are both one-to-one and onto.

  12. f:A→B.We say that the function is injective if for all x₁,x₂ of A ,x₁≠x₂,we have f(x₁)≠f(x₂). f:A→B is a function : a) f is an injective function b)if we have x₁,x₂ of A so f(x₁)=f(x₂),then x₁=x₂.

  13. f:A→B. We say that the function is surjective if for all y of B it exists x of A so f(x)=y. • F surjective↔Imf=B. • f:A→B is bijectiveif f is injective and bijective in the same time.

  14. INVERSE FUNCTION An inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa, ƒ(x)=y, and g(y)=x. More directly, g(ƒ(x))=x, meaning g(x) composed with ƒ(x) leaves x unchanged. The function g is named the inverse of the function f and it is noted g=f⁻¹: f(x)=y↔f⁻¹(y)=x,for all x of A.

  15. Monotonic functions • A function f:A→R is monotonic on A if f is increasing or decreasing on A.We say that f is strictly monotonic on A if f is strictly increasing or strictly decreasing on A. • EXAMPLES: f:R→R, f(x)=│x│ • For x₁,x₂ of (-∞,0),x₁≠x₂, f(x₁)-f(x₂)/x₁-x₂=-1→f is strictly decreasing • For x₁,x₂ of [0,-∞),x₁≠x₂, f(x₁)-f(x₂)/x₁-x₂=1→f is strictly increasing.

  16. Variation table x +∞ 0 -∞ +∞ -∞ f(x) 0

  17. Recap • 1) Function is defined by f(x)=3x2-7x-5. Find f(x-2). • 2)Find the range of f(x)=Ix-2I+3. • 3)Functions f and g are defined by f(x)=-7x-5 and g(x)=3/2x-12.Find f(g(x)). • 4)Give the intervals where the function f(x)=2x-4 is increasing and decreasing. • 5)Prove that the function f(x)=x2 is bijective!

  18. Bibliography • Manual of MATH IN ENGLISH. • Manual of MATH 10th grade • Notebooks 10th grade.

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