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Questions and Answers
What is a key characteristic of a function from set A to set B?
What is a key characteristic of a function from set A to set B?
Which of the following statements correctly describes the domain and codomain of a function?
Which of the following statements correctly describes the domain and codomain of a function?
In the function f: A → B, if A = {a, b, c} and B = {1, 2, 3, 4}, which of the following subsets represents a valid function?
In the function f: A → B, if A = {a, b, c} and B = {1, 2, 3, 4}, which of the following subsets represents a valid function?
What does the notation f(x) typically represent in relation to a function?
What does the notation f(x) typically represent in relation to a function?
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When determining if a formula defines a function, what aspect must be clarified first?
When determining if a formula defines a function, what aspect must be clarified first?
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Which of the following pairs could NOT represent a function from set A to set B?
Which of the following pairs could NOT represent a function from set A to set B?
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What is true about the outputs of a function with an infinite domain?
What is true about the outputs of a function with an infinite domain?
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Which statement accurately describes the function notation f: A → B?
Which statement accurately describes the function notation f: A → B?
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For two functions to compose, what must be true about their codomains and domains?
For two functions to compose, what must be true about their codomains and domains?
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Which of the following statements regarding composition of functions is true?
Which of the following statements regarding composition of functions is true?
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Given f(x) = x² and g(x) = x + 3, what is the output of g ∘ f for the input x?
Given f(x) = x² and g(x) = x + 3, what is the output of g ∘ f for the input x?
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Which of the following compositions is not defined based on the given functions k(x) = sin(x) and h(x) = ln(x)?
Which of the following compositions is not defined based on the given functions k(x) = sin(x) and h(x) = ln(x)?
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What can be stated if both g is a right inverse and a left inverse of f?
What can be stated if both g is a right inverse and a left inverse of f?
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The operation of functions is said to be associative. Which statement correctly reflects this property?
The operation of functions is said to be associative. Which statement correctly reflects this property?
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What conclusion can be drawn if g(f(x)) = 1_B?
What conclusion can be drawn if g(f(x)) = 1_B?
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Which one of these functions represents a scenario where composition is valid?
Which one of these functions represents a scenario where composition is valid?
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In function composition, which of the following is NOT a consequence of the identity law?
In function composition, which of the following is NOT a consequence of the identity law?
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What are the three essential components of a function?
What are the three essential components of a function?
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When is a function considered variable?
When is a function considered variable?
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For the function defined as 𝑓(𝑥) = { 𝑥, 𝑖𝑓 𝑥 ≥ 0; −1, 𝑖𝑓 𝑥 < 0, what is the value of 𝑓(−2)?
For the function defined as 𝑓(𝑥) = { 𝑥, 𝑖𝑓 𝑥 ≥ 0; −1, 𝑖𝑓 𝑥 < 0, what is the value of 𝑓(−2)?
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What does it mean for two functions 𝑓 and 𝑔 to be equal?
What does it mean for two functions 𝑓 and 𝑔 to be equal?
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Which statement is true about a constant function?
Which statement is true about a constant function?
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What is the identity map on set 𝐴?
What is the identity map on set 𝐴?
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When restricting a function 𝑓: 𝐴 → 𝐵 to a subset 𝑆, what is this new function called?
When restricting a function 𝑓: 𝐴 → 𝐵 to a subset 𝑆, what is this new function called?
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How can one define the projection map from 𝐴 × 𝐵?
How can one define the projection map from 𝐴 × 𝐵?
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Which function represents a case where 𝐵 is constant across all inputs in 𝐴?
Which function represents a case where 𝐵 is constant across all inputs in 𝐴?
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What characterizes the inclusion map from 𝑆 to 𝐴?
What characterizes the inclusion map from 𝑆 to 𝐴?
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What is a necessary condition for functions 𝑓 and 𝑔 to be considered equal?
What is a necessary condition for functions 𝑓 and 𝑔 to be considered equal?
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For the function defined by 𝑓: ℝ → ℝ where 𝑓(𝑥) = 𝑥 if 𝑥 ≥ 0 and −1 if 𝑥 < 0, what is true for 𝑓(0)?
For the function defined by 𝑓: ℝ → ℝ where 𝑓(𝑥) = 𝑥 if 𝑥 ≥ 0 and −1 if 𝑥 < 0, what is true for 𝑓(0)?
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What does it mean when a function is defined on sets 𝐴 and 𝐵?
What does it mean when a function is defined on sets 𝐴 and 𝐵?
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What is the image of a set $P$ under the function $f$?
What is the image of a set $P$ under the function $f$?
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Which of the following correctly describes the inverse image of a set $Q$ under the function $f$?
Which of the following correctly describes the inverse image of a set $Q$ under the function $f$?
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If $f(x) = x^2$, what is $f(-2)$?
If $f(x) = x^2$, what is $f(-2)$?
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Which of the following statements is true regarding the image of $ ext{P}$ under $f$ and $ ext{Q}$ under $f^{-1}$?
Which of the following statements is true regarding the image of $ ext{P}$ under $f$ and $ ext{Q}$ under $f^{-1}$?
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In Example 4.2.2, what is $f( ext{2})$ considered?
In Example 4.2.2, what is $f( ext{2})$ considered?
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What does the notation $f^{-1}( ext{Q})$ signify?
What does the notation $f^{-1}( ext{Q})$ signify?
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What is the output of $f^{-1}( ext{{1}})$ if $f(x) = x^2$?
What is the output of $f^{-1}( ext{{1}})$ if $f(x) = x^2$?
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What conclusion can be drawn if $C
subseteq D$?
What conclusion can be drawn if $C subseteq D$?
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If $S
subseteq T$, what can be inferred about $f^{-1}(S)$ and $f^{-1}(T)$?
If $S subseteq T$, what can be inferred about $f^{-1}(S)$ and $f^{-1}(T)$?
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What does the term 'composition of functions' refer to?
What does the term 'composition of functions' refer to?
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If $f(x) = x^2 + 3$, what is the result of $f(f(0))$?
If $f(x) = x^2 + 3$, what is the result of $f(f(0))$?
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What can be deduced about the functions $f: A
ightarrow B$ and $g: B
ightarrow C$?
What can be deduced about the functions $f: A ightarrow B$ and $g: B ightarrow C$?
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If two functions are inverses, what property do they share when composed?
If two functions are inverses, what property do they share when composed?
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What does the statement $f(f^{-1}(C))
eq C$ imply?
What does the statement $f(f^{-1}(C)) eq C$ imply?
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Study Notes
Fundamentals of Functions
- Functions are composed of a domain, codomain, and a subset of the product of the domain and codomain, subject to a specific condition. The domain is the set of input values, the codomain is the set of possible output values, and the subset is the set of mappings from inputs to outputs.
- A function maps each element in the domain to exactly one element (output) in the codomain.
- A function can be represented graphically by a plot of points or algebraically by an equation that links inputs (x) to outputs (y).
- Input variables are independent variables, while output variables are dependent variables.
- A function is represented by the notation f(x) = y, where f is the name of the function, x is the independent variable, and y is the dependent variable.
- Function notation follows this format, y = f(x) where x is the input (independent variable), and y is the output (dependent variable) and the name of the function.
- Not all formulas necessarily define a function. A formula needs both a domain and a codomain for it to properly define the function.
Types of Functions
- Constant maps: A function where all inputs map to a single output value (i.e., f(x) = c for all x).
- Identity maps: A function that returns the input value (i.e., f(x) = x).
- Inclusion maps: Maps a subset of one set to a larger superset.
- Restriction maps: A subset of input domain of a larger function which is restricted
- Extension maps: An extension of a function's domain.
- Projection maps: maps an element from two variables set to a single variable set (for example A x B → A).
Function Composition
- Combining two or more functions to create a new function. The output of the first function becomes the input of the second.
- Function composition is written as (g°f)(x) = g(f(x)), meaning the function g is performed after function f.
- The Codomain of the first function must equal the domain of the second function for composition to be valid.
Injectivity, Surjectivity, and Bijectivity
- Injective (one-to-one): Each distinct input has a distinct output; no two inputs map to the same output.
- Surjective (onto): Every element in the codomain is the output of at least one element in the domain.
- Bijective (one-to-one and onto): A function that is both injective and surjective.
- Graphs can be analyzed for injectivity, surjectivity and bijectivity by applying the horizontal line test, where horizontal lines intersect the graph at most once, twice, or multiple times (which determine if functions are injective, surjective or neither).
Inverse Functions
- The inverse function reverses the original function's mapping process.
- To be invertible, a function must be bijective (one-to-one and onto).
- If a function and its inverse exists, they satisfy the condition that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Defining f⁻¹
- Notations : f⁻¹(Q) = {x ∈ A | f(x) ∈ Q}, which refers to a set of x in the domain that maps to Q in the codomain This notation defines an inverse function as a set.
- A function f has an inverse f⁻¹ (which is also a function) if and only if f is bijective.
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Description
This quiz explores the essential concepts of functions, including their definition, notation, and representation both graphically and algebraically. It covers the relationships between domain, codomain, and independent and dependent variables. Test your understanding of function mapping and notation with this comprehensive quiz.
