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Questions and Answers
What is the definition of the product of two real functions f and g?
What is the definition of the product of two real functions f and g?
- The product is defined as $(fg)(x) = f(x) \div g(x)$ for all $x \in X$.
- The product is defined as $(fg)(x) = f(x) g(x)$ for all $x \in X$. (correct)
- The product is defined as $f(x) - g(x)$ for all $x \in X$.
- The product is defined as $f(x) + g(x)$ for all $x \in X$.
Which property is true for an even function?
Which property is true for an even function?
- f(x) = f(0) for all x in the domain.
- The graph of the function is symmetric about the y-axis. (correct)
- The function must have a minimum value at x = 0.
- f(-x) = -f(x) for all x in the domain.
What is required for the quotient of two functions f and g to be defined?
What is required for the quotient of two functions f and g to be defined?
- g(x) must equal 0 for all x in the domain.
- g(x) must be greater than 0 for some x in the domain.
- g(x) must not equal 0 for all x in the domain. (correct)
- f and g must be even functions.
What characterizes a periodic function?
What characterizes a periodic function?
For a function to be classified as odd, which of the following must hold true?
For a function to be classified as odd, which of the following must hold true?
What conditions must a function satisfy to be classified as one-one?
What conditions must a function satisfy to be classified as one-one?
Which of the following describes the behavior of the modulus function when $x < 0$?
Which of the following describes the behavior of the modulus function when $x < 0$?
What is the value of the signum function when $x = 0$?
What is the value of the signum function when $x = 0$?
What is the Cartesian product of two non-empty sets P and Q?
What is the Cartesian product of two non-empty sets P and Q?
Which statement about the greatest integer function is correct?
Which statement about the greatest integer function is correct?
In a relation R from set A to set B, what is the domain?
In a relation R from set A to set B, what is the domain?
Which inequality represents the triangle inequality property of the modulus function?
Which inequality represents the triangle inequality property of the modulus function?
What is the derivative of a one-one function in terms of increasing or decreasing properties?
What is the derivative of a one-one function in terms of increasing or decreasing properties?
What does the notation R -1 represent?
What does the notation R -1 represent?
Which function is defined as $f(x) = x$ for all $x$ in the real numbers?
Which function is defined as $f(x) = x$ for all $x$ in the real numbers?
Which statement accurately describes the relationship between the range of a relation and its co-domain?
Which statement accurately describes the relationship between the range of a relation and its co-domain?
Under what condition can a relation from set A to set B be classified as a function?
Under what condition can a relation from set A to set B be classified as a function?
Which of the following properties holds true for the greatest integer function?
Which of the following properties holds true for the greatest integer function?
How many total relations can be defined from a set A of size p to a set B of size q?
How many total relations can be defined from a set A of size p to a set B of size q?
What is the correct definition of the range of a relation R?
What is the correct definition of the range of a relation R?
If a relation R contains the pair (x, y), which of the following statements is true?
If a relation R contains the pair (x, y), which of the following statements is true?
What condition must hold for the denominator of a function to be considered part of its domain?
What condition must hold for the denominator of a function to be considered part of its domain?
Given the function y = f(x), what is the first step in finding its range when its domain includes all real numbers?
Given the function y = f(x), what is the first step in finding its range when its domain includes all real numbers?
Which of the following statements correctly describes injective functions?
Which of the following statements correctly describes injective functions?
When combining two functions f(x) and g(x) to find their domain, which operation must be performed if they are being added?
When combining two functions f(x) and g(x) to find their domain, which operation must be performed if they are being added?
In identifying the range of a function defined for finite domain points, what should the range consist of?
In identifying the range of a function defined for finite domain points, what should the range consist of?
Which of the following conditions must be satisfied for two functions f and g to be considered equal?
Which of the following conditions must be satisfied for two functions f and g to be considered equal?
What must be true about the inputs of a surjective function?
What must be true about the inputs of a surjective function?
How can the range of a function defined on a finite interval be determined?
How can the range of a function defined on a finite interval be determined?
What is the range of the exponential function $f(x) = a^x$, where $a > 0$ and $a \neq 1$?
What is the range of the exponential function $f(x) = a^x$, where $a > 0$ and $a \neq 1$?
Which property of logarithms states that $\log_a(MN) = \log_a M + \log_a N$?
Which property of logarithms states that $\log_a(MN) = \log_a M + \log_a N$?
For which base $a$, where $a > 0$ and $a \neq 1$, does the logarithm function exhibit decreasing behavior?
For which base $a$, where $a > 0$ and $a \neq 1$, does the logarithm function exhibit decreasing behavior?
If $x \in (0, ∞)$ and $p$ is a positive constant, what does it imply if $\log_a x < p$ when $a > 1$?
If $x \in (0, ∞)$ and $p$ is a positive constant, what does it imply if $\log_a x < p$ when $a > 1$?
According to the properties of monotonicity of logarithms, if $a > 1$ and $0 < x < y$, which of the following inequalities holds true?
According to the properties of monotonicity of logarithms, if $a > 1$ and $0 < x < y$, which of the following inequalities holds true?
What is the value of $\log_a a$ for any positive base $a \neq 1$?
What is the value of $\log_a a$ for any positive base $a \neq 1$?
Which assessment correctly describes the behavior of the logarithm function when $x$ approaches 0 from the right (i.e., $x \to 0^+$) for $a > 1$?
Which assessment correctly describes the behavior of the logarithm function when $x$ approaches 0 from the right (i.e., $x \to 0^+$) for $a > 1$?
What is the correct expression for the fractional part of a number $x$ when $1 \leq x < 2$?
What is the correct expression for the fractional part of a number $x$ when $1 \leq x < 2$?
What is the period of the function $sin(nx)$ when $n$ is an even integer?
What is the period of the function $sin(nx)$ when $n$ is an even integer?
If $f(x)$ has a period $T$, what will be the period of $kf(cx + d)$ where $k$, $c$, and $d$ are constants?
If $f(x)$ has a period $T$, what will be the period of $kf(cx + d)$ where $k$, $c$, and $d$ are constants?
Which of the following functions does not possess a period?
Which of the following functions does not possess a period?
For which pair of functions would the period of their sum be the least common multiple (LCM) of their respective periods?
For which pair of functions would the period of their sum be the least common multiple (LCM) of their respective periods?
What is the period of the function $tan(nx)$ when $n$ is an odd integer?
What is the period of the function $tan(nx)$ when $n$ is an odd integer?
If two functions $f_1(x)$ and $f_2(x)$ have periods $T_1$ and $T_2$, which of the following describes the period of $h(x) = af_1(x) - bf_2(x)$?
If two functions $f_1(x)$ and $f_2(x)$ have periods $T_1$ and $T_2$, which of the following describes the period of $h(x) = af_1(x) - bf_2(x)$?
Which of the following statements regarding periodic functions is true?
Which of the following statements regarding periodic functions is true?
Which of the following conditions applies to the least common multiple (LCM) of rational and irrational numbers?
Which of the following conditions applies to the least common multiple (LCM) of rational and irrational numbers?
Flashcards
Cartesian Product
Cartesian Product
The set of all possible pairs formed by taking one element from each set, with the order of elements maintained. For sets A and B, it's represented as A × B, and consists of pairs (a, b) where 'a' belongs to A and 'b' belongs to B.
Relation
Relation
Any subset of the Cartesian product of two sets A and B. It's a way to link elements of A to B, with each element in A potentially having corresponding elements in B.
Domain of a Relation
Domain of a Relation
The set containing all the first elements (the 'x' values) of the ordered pairs within a relation R. It denotes the inputs of the relation.
Range of a Relation
Range of a Relation
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Function
Function
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Domain of a Function
Domain of a Function
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Range of a Function
Range of a Function
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Co-domain of a Function
Co-domain of a Function
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Even Root Rule in Domain
Even Root Rule in Domain
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Denominator Rule in Domain
Denominator Rule in Domain
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Logarithmic Function Rule
Logarithmic Function Rule
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Domain of Combined Functions
Domain of Combined Functions
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Domain of Division of Functions
Domain of Division of Functions
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Injective Function
Injective Function
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One-to-One Function
One-to-One Function
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One-to-One Function (Graphical Representation)
One-to-One Function (Graphical Representation)
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One-to-One Function (Calculus)
One-to-One Function (Calculus)
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Identity Function
Identity Function
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Constant Function
Constant Function
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Modulus Function
Modulus Function
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Signum Function
Signum Function
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Greatest Integer Function
Greatest Integer Function
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Even Function
Even Function
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Odd Function
Odd Function
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Periodic Function
Periodic Function
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Multiplication of a function by a scalar
Multiplication of a function by a scalar
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Multiplication of two functions
Multiplication of two functions
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Greatest Integer Function ( [x] )
Greatest Integer Function ( [x] )
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Fractional Part of x ( {x} )
Fractional Part of x ( {x} )
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Exponential Function
Exponential Function
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Logarithm Function
Logarithm Function
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Logarithm Product Rule
Logarithm Product Rule
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Logarithm Quotient Rule
Logarithm Quotient Rule
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Logarithm Power Rule
Logarithm Power Rule
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Logarithm of Base to Itself
Logarithm of Base to Itself
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Period of a Function
Period of a Function
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Property of Periodic Functions
Property of Periodic Functions
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Shifted Periodic Functions
Shifted Periodic Functions
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Scaling Periodic Functions
Scaling Periodic Functions
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Vertical Translation
Vertical Translation
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Combined Periodic Functions
Combined Periodic Functions
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Least Common Multiple (LCM)
Least Common Multiple (LCM)
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Study Notes
Relations and Functions - Class 11 Mathematics
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Introduction: This chapter explores linking pairs of objects from multiple sets, classifying relations as functions, and examining different types of functions.
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Relations: A relationship between two sets. Any subset of a Cartesian product (A x B) constitutes a relation.
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Cartesian Product: The set of all ordered pairs (p, q) where p belongs to set P and q belongs to set Q. Written as P x Q.
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Definition: A subset 'R' of A x B. If (a, b) ∊ R, then a is related to b by R, and b is the image of a under R.
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Domain: The set of first elements in R. (x: (x,y) ∊ R)
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Range: The set of second elements in R. (y: (x,y) ∊ R)
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Co-domain: Set B is the co-domain of the relation R.
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Number of Relations: If n(A) = p and n(B)=q, then the total number of relations that can be defined from A to B is 2pq.
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Inverse Relation: The inverse of a relation R from A to B, noted as R⁻¹, is a relation from B to A where: R⁻¹ = {(b, a): (a, b) ∊ R}.
- The domain of R is the range of R⁻¹ and the range of R is the domain of R⁻¹.
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Functions: A relation from A to B where every element in A has exactly one image in B.
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Definition: Every element in the domain (set A) has one and only one image (output) in the co-domain (set B).
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Domain, Co-domain and Range of a function: The domain is the set of possible input values, the co-domain is the set of possible output values (a larger set, or potentially equal to the range), and the range is the set of actual output values achieved by the function. When provided in the form y = f(x). The domain is the largest set of x-values for which the formula provides real y-values.
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Rules for Finding Domains:
- Even Roots: Non-negative expressions are necessary.
- Denominators: Cannot be equal to zero.
- Logarithms: Input (x) must be greater than zero, and the base (a) must be greater than zero and not equal to one.
- Combining Functions: If f(x) and g(x) have domains D₁ and D₂, the domain of f(x) + g(x) or f(x)g(x) is D₁ ∩ D₂. The domain of f(x) / g(x) is D₁ ∩ D₂ - {x: g(x) = 0}.
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Function Properties: Including one-to-one, onto, and their combination in bijective functions, based on the relations between input (domain) and output (range).
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Graphical and Calculus Methods: Use graphs to determine if relations are functions ("vertical line test") and use calculus methods to find increasing or decreasing behavior.
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Standard Function Types and Graphs: Identity, constant, modulus, signum, greatest integer, and exponential functions (including graphs and key properties).
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Properties of Modulus Function:
√x² = |x|for any real numberx|xy| = |x| |y|
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Periodic Functions:
- Definition: f(x + T) = f(x) for the positive real number T, called fundamental period (the smallest T that satisfies the equation).
- Graphical Identification: Periodic functions repeat their pattern within a certain interval of values.
- Properties of Periodic Functions: The properties of a periodic function are affected by the addition or removal of constants, multiplication by scalars, and sums or differences of periodic functions. The periods will be related to the LCM of these constants.
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Algebra of Real Functions: Operations like addition, subtraction, multiplication, and division of real functions, along with properties involving coefficients.
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Even and Odd Functions: Key definitions and graph characteristics for even functions (symmetric about the y-axis); odd functions (symmetric about the origin).
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Description
Test your knowledge on the definitions and properties of various types of functions. This quiz covers even and odd functions, periodic functions, and the characteristics of one-one functions. Assess your understanding of relations, derivatives, and modulus functions as well.
