Describe the function their types examples

This summarizes the function into more than 20 slides and it shows them with their example that typed their graphs

1. Author: ah**[email protected] Created with Pi Functions: An Overview

2. CONTENTS 1. History 3. Introduction 5. Diagrams & Expressions 2. Mathematicians Linked to Functions 4. Types of Functions 6. Summary

3. 1. History Gottfried Leibniz (1646-1716) first used "function" in 1673 for curve quantities. Leonhard Euler (1707-1783) formalized f(x) notation in 1734. Peter Dirichlet (1805-1859) gave the rigorous modern definition in 1837 as correspondence between sets. Georg Cantor (1845-1918) extended it via set theory. Isaac Newton (1643-1727) and Leibniz used functions in calculus for rates and areas.

4. 3. Introduction Definition: A function f: A to B assigns each element in A exactly one element in B. Key: Each input has EXACTLY ONE output (unlike relations). Domain: Set A (all inputs). Codomain: Set B (all possible outputs). Range: Actual outputs (subset of B). Example: A=(1,2,3), B=(a,b,c,d), f=(1,a),(2,c),(3,a). Domain=(1,2,3), Range=(a,c). Function vs Relation: NOT Function: R=(1,x),(1,y) - 1 maps to two! IS Function: f=(1,x),(2,y) - each maps once.

5. 4. Types of Functions One-to-One (Injective) Different inputs give different outputs. Ex: f=(1,a),(2,b),(3,c). Onto (Surjective) Every codomain element is used. Ex: f=(1,x),(2,y),(3,x). Bijective Both one-one and onto. Ex: f=(1,a),(2,b),(3,c). Identity f(x)=x, maps to itself. Ex: I=(1,1),(2,2),(3,3). Constant All map to one value. Ex: f(x)=5 for all x. Polynomial f(x)=ax squared+bx+c. Ex: f(x)=2x squared-3x+1. Rational f(x)=p(x)/q(x), q not 0. Ex: f(x)=1/(x-2). Modulus f(x)=|x|, always positive. Ex: |-5|=5, |3|=3. Signum 1 if x greater 0, -1 if less, 0 if zero. Ex: sgn(5)=1, sgn(-3)=-1. Greatest Integer Floor function, largest int below. Ex: floor 3.7=3, floor -2.3=-3.

6. 5. Diagrams & Expressions Arrow Diagram Example: f: (1,2,3) to (a,b,c). Domain: 1, 2, 3 arrows to Codomain: a, b, c.

7. Vertical Line Test If any vertical line cuts the graph more than once, it is NOT a function. This ensures one output per input.

8. Graph Types Linear f(x)=mx+c (straight line). Cubic f(x)=x cubed (S-curve). Quadratic f(x)=ax squared+bx+c (parabola). Reciprocal f(x)=1/x (hyperbola).

9. Key Formulas Composition (gof)(x)=g(f(x)). Ex: f(x)=2x, g(x)=x+3, then gof(x)=2x+3. 1 Even/Odd Even: f(-x)=f(x) (x squared). Odd: f(-x)=-f(x) (x cubed). 3 Inverse f inverse exists if bijective. f(f inverse(x))=x. Ex: f(x)=2x+1, f inverse=(x-1)/2. Periodic f(x+T)=f(x). Ex: sin(x) period 2pi. 2 4

10. Domain Rules Polynomial all real numbers. Rational exclude denominator zeros. Root expression under root greater or equal to 0.

11. Properties 1 Composition associative: (fog)oh=fo(goh). 4 2 If f,g a NOT commutative: fog not equal to gof. 3 Identity: foI=Iof=f.