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Questions and Answers
What is the output of the function f when the input is 2, given that f(x) = 6x + 28?
What is the output of the function f when the input is 2, given that f(x) = 6x + 28?
- 36
- 44
- 28
- 40 (correct)
What does the floor function ⌊x⌋ return when x is a real number?
What does the floor function ⌊x⌋ return when x is a real number?
- The smallest integer greater than x
- The next integer after x
- The largest integer less than or equal to x (correct)
- The integer part of x without rounding
Which function defines g(n) based on whether n is odd or even?
Which function defines g(n) based on whether n is odd or even?
- g(n) = n + 1
- g(n) = 3n + 4
- g(n) = 2n + 3
- g(n) = 2n or 3n^2 + n + 1 (correct)
For the function â„“(E) defined for healthy African elephants, what does â„“(E) represent?
For the function â„“(E) defined for healthy African elephants, what does â„“(E) represent?
What is the output of g(4), given that g(n) is defined for odd and even n?
What is the output of g(4), given that g(n) is defined for odd and even n?
What is the result of applying the floor function to the number -32?
What is the result of applying the floor function to the number -32?
If n is odd and g(n) = 2n, what is g(3)?
If n is odd and g(n) = 2n, what is g(3)?
If the output of f(0) is 28, what term in the formula contributes to this result?
If the output of f(0) is 28, what term in the formula contributes to this result?
What does the ceiling function ⌈x⌉ return for a real number x?
What does the ceiling function ⌈x⌉ return for a real number x?
If x = -54, what is the value of the ceiling function ⌈x⌉?
If x = -54, what is the value of the ceiling function ⌈x⌉?
Which of the following correctly describes arity in functions?
Which of the following correctly describes arity in functions?
What term is used to describe a function with two arguments?
What term is used to describe a function with two arguments?
What is the definition of a function from set A to set B?
What is the definition of a function from set A to set B?
Given the binary function f(x,y) = x + y, which of the following is an equivalent expression?
Given the binary function f(x,y) = x + y, which of the following is an equivalent expression?
Which of the following represents the proper notation of a function?
Which of the following represents the proper notation of a function?
How can a tuple be thought of in terms of functions?
How can a tuple be thought of in terms of functions?
In the context of functions, what can be said about the domain?
In the context of functions, what can be said about the domain?
For the infinite sequence (b0, b1, b2,...), how is it represented as a function?
For the infinite sequence (b0, b1, b2,...), how is it represented as a function?
Why is the relation f(x) as 'parent of x' not considered a function?
Why is the relation f(x) as 'parent of x' not considered a function?
Which of the following is true about the function f(x) = 'mother of x'?
Which of the following is true about the function f(x) = 'mother of x'?
What is the relationship between the ceiling function and the floor function?
What is the relationship between the ceiling function and the floor function?
What characterizes the function f(x) = 'set of all children of x'?
What characterizes the function f(x) = 'set of all children of x'?
If you define f : A → B where A = {a, b, c} and B = {1, 2, 3}, what could be a possible description of the function?
If you define f : A → B where A = {a, b, c} and B = {1, 2, 3}, what could be a possible description of the function?
How many distinct outputs can a single input have in a valid function?
How many distinct outputs can a single input have in a valid function?
What is the relationship between the set of natural numbers N and its power set P(N)?
What is the relationship between the set of natural numbers N and its power set P(N)?
What can be concluded from the set Df defined in the content?
What can be concluded from the set Df defined in the content?
What does the existence of the subset Df imply about any function f: N → P(N)?
What does the existence of the subset Df imply about any function f: N → P(N)?
If n ∈ Df, which of the following must be true?
If n ∈ Df, which of the following must be true?
What does the term 'onto' refer to in the context of functions?
What does the term 'onto' refer to in the context of functions?
What is the recurrence relation for the sequence defined in the basis step?
What is the recurrence relation for the sequence defined in the basis step?
Which of the following correctly represents the composition of the functions f and g?
Which of the following correctly represents the composition of the functions f and g?
For the functions f and g defined where g: X → X and f: X → Y, which outcome is correct for (f ◦ g)(c)?
For the functions f and g defined where g: X → X and f: X → Y, which outcome is correct for (f ◦ g)(c)?
What happens when trying to compute (f â—¦ f) or (g â—¦ f)?
What happens when trying to compute (f â—¦ f) or (g â—¦ f)?
Given the functions f(x) = 2x + 3 and g(x) = 3x + 2, what is the result of (g â—¦ f)(x)?
Given the functions f(x) = 2x + 3 and g(x) = 3x + 2, what is the result of (g â—¦ f)(x)?
Which sequence does the Fibonacci function f(n) follow based on the provided pattern?
Which sequence does the Fibonacci function f(n) follow based on the provided pattern?
In the function definitions, which statement about the codomain and domain is correct?
In the function definitions, which statement about the codomain and domain is correct?
Given the recursive definition, what is the value of f(5)?
Given the recursive definition, what is the value of f(5)?
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Study Notes
Functions
- A function from set A to set B associates each element of A with exactly one element of B
- If f associates x ∈ A with y ∈ B, we write f(x) = y, meaning "f of x is y"
- f: A → B, where A is the domain and B is the codomain
- Every element in the domain must have a corresponding element in the codomain, but not every element in the codomain needs to be mapped
Describing Functions
- Functions can be described by listing associations, drawing points and arrows, or using a graph
- Functions can be defined using formulas, case distinctions, or by non-formulaic rules
Useful Functions
- The floor function, ⌊ ⌋ : R → Z, gives the largest integer less than or equal to its input
- The ceiling function, ⌈ ⌉ : R → Z, gives the smallest integer greater than or equal to its input
Multiple Argument Functions
- Functions can have multiple arguments, denoted by f(x1,...,xn)
- Binary functions have two arguments and can use infix notation (xf y = z)
Tuples and Sequences as Functions
- A tuple (a1, a2,..., an) can be seen as a function mapping {0, 1,...,n-1} to the elements of the tuple
- An infinite sequence (b0, b1,...) can be seen as a function mapping N to the sequence elements
Function Composition
- The composition of g: A → B and f: B → C is f ◦ g: A → C, defined by (f ◦ g)(a) = f(g(a))
- Composition is only defined when the codomain of g matches the domain of f
Countability
- A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers
- The set of natural numbers itself is countable
- Not all sets are countable, such as the power set P(N) – the set of all subsets of N
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