Types Of Functions
1 hr 4 min 14 Examples
- What is a function? Overview of domain, codomain, range, image, preimage
- Find domain, codomain, range and determine if the graph is a function using the vertical line test (Examples #1-5)
- Determine if the mapping diagram is a function (Examples #5-8)
- Overview of Identity function, real-valued and integer-valued, and sum-product functions (Examples #9-10)
- Composite functions for equations and sets (Example #11-12)
- Overview of Floor functions and Ceiling functions
- Evaluate the following floor and ceiling functions (Examples #13-14)
Injective
1 hr 13 min 17 Examples
- What is a Well-Defined Function? (Examples #1-5)
- Overview of Injective Functions
- Prove or disprove the function is injective (Examples #6-10)
- Determine if the congruence modulo is injective (Examples #11-13)
- Construct an injective function (Example #14)
- Use calculus to determine if a function is one-to-one (Examples #15-17)
Surjective
51 min 12 Examples
- What is a Surjective function?
- Determine if the function is surjective (Examples #1-3)
- Determine if the function is onto given a graph (Examples #4-7)
- Construct a function that is onto (Example #8)
- Prove the function is a onto (Examples #9-11)
- Prove g(x)=f(2x) is a surjection if f(x) is onto (Example #12)
Bijection
1 hr 11 min 9 Examples
- What is a one-to-one-correspondence? (Example #1)
- Determine domain, codomain, range, well-defined, injective, surjective, bijective (Examples #2-3)
- Bijection and Inverse Theorems
- Determine if the function is bijective and if so find its inverse (Examples #4-5)
- Identify conditions so that g(f(x))=f(g(x)) (Example #6)
- Find the domain for the given inverse function (Example #7)
- Prove one-to-one correspondence and find inverse (Examples #8-9)
Asymptotic Notation
1 hr 20 min 15 Examples
- Introduction to Video: Asymptotic Notation
- What is the growth of a function?
- Overview of Big-O Notation
- Overview of Big-Omega Notation
- Overview of Big-Theta Notation
- Properties and Complexity Classes of Big-Oh
- The Asymptotic Limit Theorem to determine the growth of a function
- Given g(x)=x, determine if f(x) is big-o, big-omega, or big-theta (Examples #1-6)
- Given g(x)=x^2, determine if f(x) is big-o, big-omega, or big-theta (Examples #7-12)
- Given g(n)=logn, determine if f(n) is big-o, big-omega, or big-theta (Examples #13-14)
- Arrange the functions so they are big-o of the next function (Example #15)
Big O
1 hr 42 min 15 Examples
- Introduction to Video: Big O, Big Omega, and Big Theta
- How to prove Big-oh, Big-Omega, and Big-Theta
- Show the function is big O or big omega and find constants c and k (Examples #1-4)
- Show the function is big theta and find witnesses (Examples #5-6)
- Verify the function is big O and find witnesses (Examples #7-9)
- Verify the series is big O and find constants (Example #10)
- Find a good big-O estimate and find constants c and k (Examples #11-12)
- Find a good big-O estimate and find witnesses c and k (Examples #13-15)
Chapter Test
1 hr 0 min 7 Practice Problems
- True/False: is the function bijective (Problem #1a-b)
- Determine the number of well-defined, onto functions (Problem #2a-b)
- Evaluate the ceiling function and inverse function (Problem #3a-b_
- Find the composite function (Problem #4)
- Is the floor function one-to-one or onto. Justify (Problem #5)
- Prove the function is a bijection (Problem #6)
- Use calculus to determine if the function is one-to-one or onto (Problem #7)
