Functions and Their Properties Quiz

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

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?

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?

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?

There exists a positive real number T such that f(x + T) = f(x) for all x in R. Signup and view all the answers

For a function to be classified as odd, which of the following must hold true?

f(x) = -f(-x) for all x in the domain. Signup and view all the answers

What conditions must a function satisfy to be classified as one-one?

It must be always increasing or always decreasing. Signup and view all the answers

Which of the following describes the behavior of the modulus function when $x < 0$?

It returns the absolute value of x. Signup and view all the answers

What is the value of the signum function when $x = 0$?

0 Signup and view all the answers

What is the Cartesian product of two non-empty sets P and Q?

The set of all ordered pairs of elements from P and Q Signup and view all the answers

Which statement about the greatest integer function is correct?

It rounds down to the nearest integer. Signup and view all the answers

In a relation R from set A to set B, what is the domain?

The set of first elements in R Signup and view all the answers

Which inequality represents the triangle inequality property of the modulus function?

|x + y| ≤ |x| + |y| Signup and view all the answers

What is the derivative of a one-one function in terms of increasing or decreasing properties?

f'(x) > 0 for all x in domain. Signup and view all the answers

What does the notation R -1 represent?

The relation that pairs elements of B back to A Signup and view all the answers

Which function is defined as $f(x) = x$ for all $x$ in the real numbers?

Identity function Signup and view all the answers

Which statement accurately describes the relationship between the range of a relation and its co-domain?

The range is a subset of the co-domain Signup and view all the answers

Under what condition can a relation from set A to set B be classified as a function?

Every element of A has a unique image in B Signup and view all the answers

Which of the following properties holds true for the greatest integer function?

It satisfies $[x + n] = [x] + n$ for any integer n. Signup and view all the answers

How many total relations can be defined from a set A of size p to a set B of size q?

$2^{pq}$ Signup and view all the answers

What is the correct definition of the range of a relation R?

The set of all second elements in R Signup and view all the answers

If a relation R contains the pair (x, y), which of the following statements is true?

y is the image of x under relation R Signup and view all the answers

What condition must hold for the denominator of a function to be considered part of its domain?

It cannot be zero. Signup and view all the answers

Given the function y = f(x), what is the first step in finding its range when its domain includes all real numbers?

Determine the domain of f(x). Signup and view all the answers

Which of the following statements correctly describes injective functions?

They associate every element of the domain with a unique element of the co-domain. Signup and view all the answers

When combining two functions f(x) and g(x) to find their domain, which operation must be performed if they are being added?

D1 ∩ D2 Signup and view all the answers

In identifying the range of a function defined for finite domain points, what should the range consist of?

Corresponding f(x) values at finite points. Signup and view all the answers

Which of the following conditions must be satisfied for two functions f and g to be considered equal?

Both functions must have the same domain and co-domain. Signup and view all the answers

What must be true about the inputs of a surjective function?

All elements of the co-domain must receive an image. Signup and view all the answers

How can the range of a function defined on a finite interval be determined?

By calculating the least and greatest values of the function over that interval. Signup and view all the answers

What is the range of the exponential function $f(x) = a^x$, where $a > 0$ and $a \neq 1$?

(0, ∞) Signup and view all the answers

Which property of logarithms states that $\log_a(MN) = \log_a M + \log_a N$?

Product Property Signup and view all the answers

For which base $a$, where $a > 0$ and $a \neq 1$, does the logarithm function exhibit decreasing behavior?

0 < a < 1 Signup and view all the answers

If $x \in (0, ∞)$ and $p$ is a positive constant, what does it imply if $\log_a x < p$ when $a > 1$?

x < a^p Signup and view all the answers

According to the properties of monotonicity of logarithms, if $a > 1$ and $0 < x < y$, which of the following inequalities holds true?

$\log_a x < \log_a y$ Signup and view all the answers

What is the value of $\log_a a$ for any positive base $a \neq 1$?

1 Signup and view all the answers

Which assessment correctly describes the behavior of the logarithm function when $x$ approaches 0 from the right (i.e., $x \to 0^+$) for $a > 1$?

It approaches negative infinity. Signup and view all the answers

What is the correct expression for the fractional part of a number $x$ when $1 \leq x < 2$?

${x} = x - 1$ Signup and view all the answers

What is the period of the function $sin(nx)$ when $n$ is an even integer?

$ heta$ Signup and view all the answers

If $f(x)$ has a period $T$, what will be the period of $kf(cx + d)$ where $k$, $c$, and $d$ are constants?

$rac{T}{c}$ Signup and view all the answers

Which of the following functions does not possess a period?

$x^3 + 5$ Signup and view all the answers

For which pair of functions would the period of their sum be the least common multiple (LCM) of their respective periods?

$sin(2x)$ and $cos(4x)$ Signup and view all the answers

What is the period of the function $tan(nx)$ when $n$ is an odd integer?

2 Signup and view all the answers

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 $h(x) = af_1(x) - bf_2(x)$?

LCM of $T_1$ and $T_2$ Signup and view all the answers

Which of the following statements regarding periodic functions is true?

Adding a constant to a periodic function does not affect its period. Signup and view all the answers

Which of the following conditions applies to the least common multiple (LCM) of rational and irrational numbers?

LCM of rational and irrational numbers never exists. Signup and view all the answers

Study Notes

Relations and Functions - Class 11 Mathematics

Introduction: This chapter explores linking pairs of objects from multiple sets, classifying relations as functions, and examining different types of functions.
Relations: A relationship between two sets. Any subset of a Cartesian product (A x B) constitutes a relation.
- 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.
- 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.
- Domain: The set of first elements in R. (x: (x,y) ∊ R)
- Range: The set of second elements in R. (y: (x,y) ∊ R)
- Co-domain: Set B is the co-domain of the relation R.
- 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 2^pq.
- 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.
Functions: A relation from A to B where every element in A has exactly one image in B.
- Definition: Every element in the domain (set A) has one and only one image (output) in the co-domain (set B).
- 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.
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}.
Function Properties: Including one-to-one, onto, and their combination in bijective functions, based on the relations between input (domain) and output (range).
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.
Standard Function Types and Graphs: Identity, constant, modulus, signum, greatest integer, and exponential functions (including graphs and key properties).
Properties of Modulus Function:
- √x² = |x| for any real number x
- |xy| = |x| |y|
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.
Algebra of Real Functions: Operations like addition, subtraction, multiplication, and division of real functions, along with properties involving coefficients.
Even and Odd Functions: Key definitions and graph characteristics for even functions (symmetric about the y-axis); odd functions (symmetric about the origin).

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

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.