per 2 Ayinehsazian, Saba

182 days ago by sayineh

#Function- An equation for which any (x) that can be plugged into the equation will yield exactly one (y) out of the equation. 
       
#For example, f(x) = x^2 + 2 
       
#A function can only have one (x), or domain, or else it is not considered a function. Basically, it is NOT a function if the (x) value is REPEATING 
       
[x,f(x)] 
        
[10, 100]
[10, 100]
#Domain- The domain of a function is the set of all possible input values (usually x), which allows the function formula to work. 
       
domain = [-10..10] [f(x) for (x) in domain] 
        
[100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64,
81, 100]
[100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
[(x,f(x)) for x in domain] 
        
[(-10, 100), (-9, 81), (-8, 64), (-7, 49), (-6, 36), (-5, 25), (-4, 16),
(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5,
25), (6, 36), (7, 49), (8, 64), (9, 81), (10, 100)]
[(-10, 100), (-9, 81), (-8, 64), (-7, 49), (-6, 36), (-5, 25), (-4, 16), (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5, 25), (6, 36), (7, 49), (8, 64), (9, 81), (10, 100)]
points([(x,f(x)) for x in domain]) 
       
#Range- The range is the set of all possible output values (usually y), which result from using the function formula. 
       
range = [-10..10] [f(x) for (x) in range] 
        
[100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64,
81, 100]
[100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
#Composition-The nesting of two or more functions to form a single new function 
       
f(x) = x + 8 f [f(x) for (x) in [-5..5]] 
        
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
g(x) = x +2 g [g(x) for (x) in [-5..5]] 
        
[-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7]
[-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7]
g(f(x)) #This is an example of a composition 
        
15
15
#Inverse-An inverse is achieved when the (x) and (y) variables are switched, and we solve for (y) in the equation. 
       
#Ex. f(x) = 3*x + 2 f 
        
x |--> x + 8
x |--> x + 8
var('x y') solve(x == 3*y+2, y) 
        
[y == 1/3*x - 2/3]
[y == 1/3*x - 2/3]
g(x) = 1/2*x - 3/2 g 
        
x |--> 1/2*x - 3/2
x |--> 1/2*x - 3/2
f(g(x)) 
        
1/2*x + 13/2
1/2*x + 13/2
g(f(x)) 
        
1/2*x + 5/2
1/2*x + 5/2
[f(x) for (x) in [-5..5]] 
        
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
[g(x) for (x) in _] 
        
[-3/2, -5/4, -1, -3/4, -1/2, -1/4, 0, 1/4, 1/2, 3/4, 1]
[-3/2, -5/4, -1, -3/4, -1/2, -1/4, 0, 1/4, 1/2, 3/4, 1]

First Semester

  • Functions

    • Definition of a function

      • Domain
      • Range
      • Composition
      • Inverse

    • Types of Functions

      • Polynomial
        • Degree
        • Leading coefficient
        • Degree of polynomial using constant difference
      • Quadratic Functions
        • Minimum and maximum
        • Roots
        • Word Problems
      • Exponential Functions
        • Graph
        • Domain
        • Range
      • Square Root
        • Domain
        • Range
        • Composition
      • Cubic functions
      • Even/odd functions
      • Ceiling/floor function
        • Evaluate
        • Word problems

    • Transformations

      • Scale factors of points
      • Translation of points
      • Translations of functions

    • Sequence and series

      • Convergence and divergence
      • Limit
      • Recursive
      • Summation notations
      • Arithmetic series
      • Sum of infinite geometric series
 
       

List comprehension is a very useful way to express mathematical ideas. It is a way of creating lists from other lists.

It has the structure [f(x) for x in [some list]] where f(x) can be a previously defined function or just some general expression:
[factorial(x) for x in [0..10]] 
       
f(x) = x^2 + 2*x - 5 [f(x) for x in [0..10]] 
       
domain = [-10..10] [f(x) for x in domain] 
       
[(x,f(x)) for x in domain] 
       
points([(x,f(x)) for x in domain]) 
       

Here's an idea for illustrating inverses:
F = [(x,f(x)) for x in domain] [(y,x) for (x,y) in F] 
       
points([(y,x) for (x,y) in F]) 
       
f(x) = 2*x + 3 f 
       
var('x y') solve(x == 2*y+3, y) 
       
g(x) = 1/2*x - 3/2 g 
       
f(g(x)) 
       
g(f(x)) 
       
[f(x) for x in [0..10]] 
       
[g(x) for x in _] 
        
[0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5]
[0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5]

    • Extra topics we've covered:

      • Factorials
      • Permutations
      • Combinations
      • Binomial Theorem
      • Binomial Probability
 
       

Answer the following with

  • A - I know it's true.
  • B - I believe it's true.
  • C - I really don't know.
  • D - I believe it's false.
  • E - I know it's false.
 
  1. \sum_{x=1}^{n}x = \frac{n(n+1)}{2}
  2. \sum_{i=1}^{\infty}\frac{1}{2^i} = 1
  3. (p + q)^3 = \sum _{r=0}^{3} (_{3}C_{r})(p^{3-r})(q^{r})
  4. (p + q)^n = \sum _{r=0}^{n} (_{n}C_{r})(p^{n-r})(q^{r})
  5. The third term in the expansion of (p + q)^n is _{n}C_{2}p^{n-2}q^{2}.
  6. \sum_{r=0} ^{n} (_{n} C _{r}) = 1
  7. If you toss 10 coins into the air, the probability that 5 of them will land heads and 5 will land tails is 50\%.
  8. If you guess on 10 questions on a True/False test, the probability that you will guess 5 right and 5 wrong is 50\%.
  9. If you guess on 10 questions on a Multiple Choice test with 4 choices per question, the probability that you will guess all of them wrong is about 6\%.
  10. If you guess on 10 questions on a Multiple Choice test with 4 choices per question, the probability that you will guess 5 right and 5 wrong is about 6\%.
  11. If you toss a die 10 times, the probability that you will get 7 fours is 35%.
  12. \sum_{k=1}^{\infty} \frac{1}{3^k} = \frac{1}{2}
  13. \sum_{k=1}^{\infty} \frac{1}{n^k} = \frac{1}{n-1}