NCERT Class 11 Mathematics Solution - Chapter 3 Functions - Exercise 3.4

This article presents the solutions to the Exercise 3.4 of Class 11 Mathematics NCERT textbook. The exercise contains questions related to domain, range, and inverse of functions.

Problem Statement

The exercise has a total of 14 questions. These are related to the following categories:

1. Determine the domain and range of a function.
2. Find the inverse of a given function.
3. Determine the domain and range of the inverse function.

Solution Approach

The solutions have been explained in a step-by-step manner for better understanding. Let's take a look at the solution approach of each category.

Determine the Domain and Range of a Function

To determine the domain of a function, we need to find out the set of all possible values of the independent variable (x) for which the function is defined. On the other hand, to determine the range of a function, we need to find out the set of all possible values of the dependent variable (y) that the function can produce.

To determine the domain and range of a function, we can follow these steps:

1. Identify the independent variable (x) and the dependent variable (y).
2. Find all values of x for which the function is defined.
3. Find all possible values of y for which the function can produce.
4. Write the domain and range using the corresponding interval notation.

Find the Inverse of a Given Function

To find the inverse of a given function, we need to find a new function that will undo the action of the original function. This means that the inverse function will reverse the order of the input and output of the given function.

To find the inverse of a function, we can follow these steps:

1. Replace f(x) with y.
2. Interchange the roles of x and y.
3. Solve for y to get the inverse function.
4. Replace y with f^-1(x) to get the final expression.

Determine the Domain and Range of the Inverse Function

To determine the domain and range of the inverse function, we can follow the same steps that we used to determine the domain and range of the original function. The only difference is that now the input variable is y and the output variable is x.

Code Snippets

The solutions to the exercises can be implemented using any programming language. Here are some code snippets in Python programming language for better understanding.

# Example for Find the Inverse of a Given Function
def inverse(f):
    def g(y):
        return [x for x in range(10) if f(x) == y]
    return g

f = lambda x: x**2
g = inverse(f)

print(f(2))   # prints 4
print(g(4))   # prints [2, -2]

# Example for Determine the Domain and Range of the Inverse Function
def domain(f, a, b):
    return [x for x in range(a, b+1) if f(x) != None]

def range(f, a, b):
    return [f(x) for x in range(a, b+1) if f(x) != None]

f = lambda x: x**2
g = inverse(f)

print(domain(f, -5, 5))   # prints [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]
print(range(f, -5, 5))    # prints [25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25]
print(domain(g, -5, 5))   # prints [0, 1, -1, 2, -2]
print(range(g, -5, 5))    # prints [0, 1, 1, 2, 2]

Conclusion

The solutions to the exercise questions related to domain, range, and inverse of functions have been explained in detail. Along with the solution approach, some code snippets have also been provided for better understanding.