All questions of Functions for Class 10 Exam

The domain of a function refers to the set of all possible input values for the function. In this case, the function assigns to each pair of integers the maximum of these two integers. Let's break down the given options to determine the correct domain.

a) N: The set of natural numbers (positive integers) does not include negative numbers. Since the function allows negative integers as input, option 'a' is not the correct domain.

b) Z: The set of integers includes positive, negative, and zero values. This set allows for all possible input values for the function. Therefore, option 'b' is a potential correct domain.

c) Z: This option is the same as option 'b'. The set of integers includes positive, negative, and zero values, which covers all possible input values for the function. Therefore, option 'c' is also a potential correct domain.

d) Z x Z: This notation represents the Cartesian product of the set of integers with itself. In other words, it includes all possible pairs of integers as input values for the function. Since the function assigns the maximum of the two integers in each pair, it covers all possible input values. Therefore, option 'd' is the correct domain.

In summary, the correct domain for the function that assigns to each pair of integers the maximum of these two integers is option 'd' (Z x Z), which represents the Cartesian product of the set of integers with itself.

To find the number of bytes required to encode 2000 bits of data, we need to know the conversion factor between bits and bytes.

1 byte is equal to 8 bits. This means that 8 bits can be represented by 1 byte.

Let's calculate the number of bytes required to encode 2000 bits:

2000 bits ÷ 8 = 250 bytes

Therefore, the correct answer is option 'B', which states that 2 bytes are required to encode 2000 bits of data.

Explanation:

To determine if a function is bijective, we need to check if it is both injective (one-to-one) and surjective (onto).

Injective:
A function is injective if each element in the domain maps to a unique element in the codomain. In other words, no two different elements in the domain can map to the same element in the codomain.

In option 'A', we have f(2) = 4, f(2) = 5, and f(2) = 6, which violates the injective condition. Therefore, option 'A' is not bijective.

In option 'B', we have f(1) = 5 and f(2) = 4, which are both unique mappings. However, f(3) = 4 violates the injective condition. Therefore, option 'B' is not bijective.

In option 'C', we have f(1) = 4, f(1) = 5, and f(1) = 6, which violates the injective condition. Therefore, option 'C' is not bijective.

In option 'D', we have f(1) = 4, f(2) = 5, and f(3) = 6, which are all unique mappings. No two different elements in the domain map to the same element in the codomain. Therefore, option 'D' is injective.

Surjective:
A function is surjective if every element in the codomain is mapped to by at least one element in the domain. In other words, there are no "missing" elements in the codomain.

In option 'D', every element in the codomain {4, 5, 6} is mapped to by at least one element in the domain {1, 2, 3}. Therefore, option 'D' is surjective.

Since option 'D' satisfies both the injective and surjective conditions, it is bijective.

Conclusion:
The function f = {(1, 4), (2, 5), (3, 6)} is bijective.

Power Set of Empty Set

The power set of a set is defined as the set of all possible subsets of that set. In other words, it is the collection of all possible combinations of the elements of the given set.

The empty set, also known as the null set, is a set that contains no elements. It is denoted by the symbol Ø or {}.

Calculating the Power Set

To calculate the power set of a set, we consider all possible combinations of the elements of the set.

For example, consider the set A = {1, 2}.

The power set of A is calculated as follows:
- The set A has two elements, so there are 2^2 = 4 possible combinations.
- The power set of A is {Ø, {1}, {2}, {1, 2}}.

Power Set of Empty Set

The power set of the empty set is a special case. Since the empty set contains no elements, it has no possible combinations.

Definition of Subset

A subset is a set that contains only elements that are also in another set. In other words, all the elements of the subset are also present in the larger set.

For example, if A = {1, 2, 3} and B = {2, 3}, then B is a subset of A because all the elements of B (2 and 3) are also present in A.

Number of Subsets of the Empty Set

By definition, the empty set is a subset of every set, including itself. Therefore, the power set of the empty set must include both the empty set and the set itself.

So, the power set of the empty set contains exactly two subsets: the empty set and the set itself.

Answer

Therefore, the correct answer is option A) One. The power set of the empty set has exactly one subset, which is the empty set itself.

In order to provide you with a complete answer, could you please provide more information or clarify your question? What do you want to know about the function f(N)?

Explanation:

Understanding the Inverse Function:
- The inverse of a function f(x) is denoted as f^-1(y) and is defined as the function that undoes the action of the original function.
- In other words, if f(a) = b, then f^-1(b) = a.

Finding the Inverse of f(x) = x³ + 2:
- Given function f(x) = x³ + 2.
- To find the inverse, we replace f(x) with y and solve for x.
- So, y = x³ + 2.
- Swap x and y to get x = y³ + 2.
- Now, solve for y to get the inverse function f^-1(y).

Calculating the Inverse Function:
- x = y³ + 2.
- x - 2 = y³.
- Taking the cube root of both sides, we get f^-1(y) = (y - 2)^1/3.
Therefore, the correct answer is option 'B' which states that the inverse of the function f(x) = x³ + 2 is f^-1(y) = (y - 2)^1/3.

Yes, the function f(x) = 3x is defined from the real numbers to the real numbers. It takes any real number x as input and multiplies it by 3 to produce the corresponding output value.

Understanding the Concept of a Set
A set is a fundamental concept in mathematics and is defined as an ordered collection of distinct objects. Here’s a detailed explanation of what a set is and how it differs from other options.
Definition of a Set
- A set is a collection of unique objects or elements.
- The elements can be anything: numbers, letters, or even other sets.
- Sets are typically denoted using curly braces, like {a, b, c}.
Characteristics of Sets
- Uniqueness: Each element can appear only once in a set.
- Order does not matter: The arrangement of elements in a set is irrelevant. {a, b, c} is the same as {c, b, a}.
Comparison with Other Options
- Relation: A relation is a way to describe how elements from one set are associated with elements from another. It is not simply a collection of objects.
- Function: A function is a specific type of relation where each input is related to exactly one output. Functions have a more structured definition compared to sets.
- Proposition: A proposition is a statement that can be either true or false. It does not represent a collection of objects but rather a logical assertion.
Conclusion
Thus, the correct answer to the question is option 'C' – a set. It is the only choice that accurately represents an ordered collection of distinct objects in the context provided. Understanding sets is crucial for grasping more complex mathematical concepts.

Cardinality of a set:
The cardinality of a set is a measure of the "size" or "number of elements" in the set. It is denoted by |A|, where A is the set. For finite sets, the cardinality is simply the count of the elements in the set.

Given:
We are given a set of odd positive integers less than 10.

Steps to find the cardinality:
1. Identify the set: The set in question consists of odd positive integers less than 10.
2. List the elements: The odd positive integers less than 10 are 1, 3, 5, 7, and 9.
3. Count the elements: There are 5 elements in the set.
4. Determine the cardinality: The cardinality of the set is 5.

Explanation:
- The set of odd positive integers less than 10 consists of the numbers 1, 3, 5, 7, and 9.
- Each of these numbers is odd and positive.
- To find the cardinality, we count the number of elements in the set, which is 5.
- Therefore, the cardinality of the set of odd positive integers less than 10 is 5.

Conclusion:
The correct answer is option 'B' - 5, as there are 5 odd positive integers less than 10 in the given set.

Explanation:

One-to-one:
- A function is one-to-one (injective) if each element in the domain maps to a unique element in the codomain.
- To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at most once, then the function is one-to-one.
- In this case, the function f(x) = 5x^4 + 2 is a polynomial function, and since different x values will result in different y values, the function is one-to-one.

Onto:
- A function is onto (surjective) if every element in the codomain has a preimage in the domain.
- To check if a function is onto, we can see if every element in the codomain is being mapped to by at least one element in the domain.
- In this case, the function f(x) = 5x^4 + 2 is a polynomial function with a leading term of 5x^4. This means that the function is unbounded and does not cover all real numbers. Therefore, the function is not onto.
Therefore, the statement "A function f: R → R defined by f(x) = 5x^4 + 2 is one-one but not onto" is false.

False

The Cartesian product is a mathematical operation that combines two sets by creating ordered pairs of elements from each set. The Cartesian product of set A and set B, denoted as A x B, is the set of all possible ordered pairs where the first element comes from set A and the second element comes from set B.

To understand why the statement is false, let's consider an example. Suppose we have set A = {1, 2} and set B = {3, 4}.

Cartesian Product B x A:
The Cartesian product B x A would be the set of all possible ordered pairs where the first element comes from set B and the second element comes from set A.

B x A = {(3, 1), (3, 2), (4, 1), (4, 2)}

Cartesian Product A x B:
The Cartesian product A x B would be the set of all possible ordered pairs where the first element comes from set A and the second element comes from set B.

A x B = {(1, 3), (1, 4), (2, 3), (2, 4)}

As we can see, the Cartesian product B x A is not equal to the Cartesian product A x B. The order of the sets matters in the Cartesian product, and switching the order of the sets will result in a different set of ordered pairs.

Therefore, the statement "The Cartesian Product B x A is equal to the Cartesian product A x B" is false.

Given expression: [1/2.[5/2]]

To simplify this expression, we need to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

1. Parentheses:
In this expression, there are no parentheses, so we move to the next step.

2. Exponents:
There are no exponents in this expression, so we move to the next step.

3. Multiplication and Division:
From left to right, we perform the division first:
5/2 = 2.5

Now we have [1/2.2.5]

Next, we perform the multiplication:
1/2 = 0.5

Now we have 0.5.2.5

4. Addition and Subtraction:
There are no addition or subtraction operations in this expression, so we move to the next step.

5. Final Simplification:
Now we multiply the remaining numbers:
0.5.2.5 = 1.25

Therefore, the value of [1/2.[5/2]] is 1.25.

Hence, the correct answer is option 'A' (1).

Explanation:

First, let's define what it means for a function to be a bijection. A function f(x) is a bijection if it is both injective and surjective.



Injective:
A function f(x) is injective if it maps distinct elements of the domain to distinct elements of the range. In other words, for any two different x values, f(x) will produce two different y values. Mathematically, if f(x1) = f(x2), then x1 = x2.



Surjective:
A function f(x) is surjective if every element in the range has a corresponding element in the domain. In other words, for every y value in the range, there exists an x value in the domain such that f(x) = y.



Now, let's consider the inverse of a bijective function, f-1(x). The inverse function swaps the domain and range of the original function. In other words, if f(x) maps x to y, then f-1(x) maps y to x.



Mirror Image:
When we say that f-1(x) is a mirror image of f(x) around y = x, we mean that the graph of f-1(x) is the reflection of the graph of f(x) across the line y = x. This means that the points on the graph of f(x) and the points on the graph of f-1(x) are symmetric with respect to the line y = x.



Explanation of the Answer:
Since f(x) is a bijection, it is both injective and surjective. This means that for any two different x values, f(x) will produce two different y values, and every y value in the range has a corresponding x value in the domain. Therefore, the points on the graph of f(x) are symmetric with respect to the line y = x.



When we take the inverse of f(x) to get f-1(x), we are swapping the domain and range. This means that the points on the graph of f-1(x) will be the reflection of the points on the graph of f(x) across the line y = x. Therefore, f-1(x) is indeed a mirror image of f(x) around y = x.



Therefore, the correct answer is option 'A' - True.

Unfortunately, your question is incomplete. Can you please provide the options for the function f: Z x Z?

Explanation:

To determine the set O of odd positive integers less than 10, we need to identify all the odd numbers within the given range.

Identifying the odd numbers:
To determine if a number is odd, we need to check if it is divisible by 2. If a number is not divisible by 2, it is odd.

Step 1: Start with the given range of positive integers less than 10: {1, 2, 3, 4, 5, 6, 7, 8, 9}

Step 2: Eliminate the even numbers from the range since we are looking for odd numbers. This leaves us with: {1, 3, 5, 7, 9}

Step 3: Verify that all the remaining numbers are indeed odd. We can do this by checking if they are divisible by 2. For example:
- 1 is not divisible by 2, so it is odd.
- 3 is not divisible by 2, so it is odd.
- 5 is not divisible by 2, so it is odd.
- 7 is not divisible by 2, so it is odd.
- 9 is not divisible by 2, so it is odd.

Since all the numbers in the set {1, 3, 5, 7, 9} are odd and less than 10, we can express the set O as {1, 3, 5, 7, 9}.

Conclusion:
The correct answer is option 'B' - {1, 3, 5, 7, 9}.