Foundation of Mathematics Dr. Sami A. Hussein

1. Iraqi Kurdistan Regional GovernmentMinistry of Higher Education & Scientific ResearchSalahaddin University - ErbilCollege of Basic EducationDepartment of MathematicsFirst class student Foundation of Mathematics Dr. Sami A. Hussein 2018-2019

2. Function • Moreover A is called domain and B is called co-domain.

3. Example f is a function a b c d 1 2 3 4 5 a b c d 1 2 3 4 5 g is a function

4. Example This is not a function a b c 1 2 3 4 5 a b c d 1 2 3 4 This is not a function

5. Examples 1. Let defined by f(x)=x2 is a function. 2. Let defined by is not a function. 3. Let defined by is not a function. 4. Let defined by is a function.

6. Vertical Line Test for Functions The graph of a function can be crossed at most once by any vertical line. Not a Function Function

7. Linear Functions: A function of the form f(x)= mx+b, for constants m and b, is called a linear function. Types of Functions • Constant functions result when the slope m = 0 • identity function is result when the slope m = 1 and • b = 0. y=x y=x

8. 2. Polynomial Functions : A function f is a polynomial if where n is a nonnegative integer and the numbers are real constants.

9. 3. Algebraic Functions: An algebraic function is a function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots).

10. Algebra of Functions Let f(x) and g(x) be to functions, then

11. 4. Trigonometric Functions: such as sine and cosine functions. 5. Exponential Functions: Functions of the form where the base a>0 is a positive constant and a≠ 1, are called exponential functions. 6. Logarithmic Functions: These are the functions , where the base a≠1 is a positive constant. 7. Transcendental Functions: These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions.

12. Identify each function given here as one of the types of functions we have discussed. Keep in mind that some functions can fall into more than one category. • Example

13. Some notes to find the domain of the function 1- If a function is a polynomial then 2- for if n be even then , if n be odd then . 3- If f(x)= h(x)/g(x) where each of h(x) and g(x) are polynomial then

14. Example Verify the domains of these functions. 1. 2. 3. 4. 5. 6.

15. Homework 1. find the domain of the following functions a. b. c. d.

16. One-to-One Functions • A function f:AB is said to be one-to-one or injective, if and only if • x1 , x2 A if x1≠x2 → f(x1) ≠ f(x2) OR if f(x1) = f(x2) → x1 = x2

17. a e i o 1 2 3 4 5 a e i o 1 2 3 4 5 Example one-to-one function A function that is not one-to-one

18. Determine whether the following functions are one-to-one: • f(x)=3x+2 2. f(x)=(x-1)2 3. 4. f(x)= sin(x) 5.

19. The Horizontal Line Test for One-to-One Functions A function is one-to-one if and only if its graph intersects each horizontal line at most once.

20. Onto Functions A function f:AB is called onto, or Surjective, if and only if f(A)=B. Bijection Functions • A function f: AB is called bijection, if and only if • f is a 1-1 function • f is an onto function.

21. a e i o u 1 2 3 4 a e i o 1 2 3 4 5 Example An onto function A function that is not onto

23. Composition of Functions The composition of the functions f(x) and g(x) is fog(x)=f(g(x)) fog “f composed by g of x equals f of g of x” C A g g f B

25. Example If f(x)= x 2 +3 and g(x)=x-1, find • (fog)(x) • (gof)(x) • (fof)(x) • (gog)(x) Homework If f(x) = x + 2 and g(x) = 4 – x2 find: (gof)(x) (fog)(4)

26. Inverse Trigonometric Functions Note : does not mean 1 /ƒ(x). B A A B a b c 1 -2 3 f 1 -2 3 a b c

27. Example