Operations on Functions MCQ Quiz in বাংলা - Objective Question with Answer for Operations on Functions - বিনামূল্যে ডাউনলোড করুন [PDF]

Calculation:

Given,

The function is such that: for all real values of x  and y , and .

We need to find the value of .

Substitute and in the given functional equation :

Since , we conclude:

Hence, the correct answer is Option 4. 

Calculation:

Given,

The function satisfies the functional equation:

Also, we are given that:

We use the functional equation to calculate f(4) . Substituting x = 4 and y = 2 , we get:

Hence, the correct answer is Option 3.

Calculation:

Given,

The function satisfies the equation for all positive real values of x and y, and .

We are tasked with finding .

Using the functional equation, for , we have:

Since , we can calculate

Next, to find , we use the functional equation again:

Since , we can calculate

Hence, the correct answer is Option 4.

Calculation:

The function is , and we need to find the area bounded by the curve and the x-axis between x = -1  and x = 1 .

The required area is given by the definite integral of the function from  -1  to 1 :

Evaluate the integral from  -1 to  1 :

∴ The area is square units.

Hence, the correct answer is Option 3.

Calculation:

Given,

The function is and we are tasked with finding .

Given , we substitute \( f(x) \) into itself:

Now, apply the function to :

This simplifies to:

Thus, .

Now, we need to find .

Substitute x = 1 into the function:

∴ The value of is -1.

The correct answer is Option (1).

Calculation:

Given,

The function is , a linear polynomial.

Since is linear with , it is one-one (injective): different give different .

The actual outputs given are , so the range of is .

The codomain is the set of natural numbers . For to be onto, its range must equal its codomain, but here:

∴ f is one-one but not onto.

Hence, the correct answer is Option 1.

Calculation:

Given,

The function is , and we are also given that the function is , where p and q  are constants.

Using the points f(1) = 1  and f(2) = 4 , we create the following system of equations:

- From f(1) = 1 , we have , so p + q = 1 .......(1).

- From f(2) = 4 , we have p(2) + q = 4 , so 2p + q = 4  ..........( 2).

Subtract Equation 1 from Equation 2:

Now, substitute p = 3 into Equation 1:

∴ The value of p + q  is 1.

The correct answer is Option (3).

Calculation:

Given, g is the inverse of f

⇒ g^-1(x) = f(x)

⇒ f(g(x)) = x

Differentiating wrt x, we get:

f '(g(x))g'(x) = 1

⇒ g'(x) = 

⇒ g'(x) = 1 + {g(x)}5 [∵ ]

∴ g'(x) is equal to 1 + {g(x)}⁵. 

The correct answer is Option 4.

Calculation

In (1) Put x = y = 0 ⇒ f(0) = 2f(0) + 1 ⇒ f(0) = –1

So, f(0) = 0 + 0 + b = –1 ⇒ b = –1

In (1) Put y = –x ⇒ f(0) = f(x) + f(–x) + 1 +  x²

–1 = 2(3a + 2)x² + 2b + 1 +  x2

⇒ 

So f(x) = 

⇒ 

Now, 

Hence option 4 is correct

Given:

f: [1.2, 1.9] → R, f(x) = [x], where [x] denotes the greatest integer less than or equal to x.

Concept Used:

The greatest integer function [x] is discontinuous at integer values.

The derivative of a constant function is 0.

Calculation:

Given:

f: [1.2, 1.9] → R, f(x) = [x], where [x] denotes the greatest integer less than or equal to x.

In the interval [1.2, 1.9), the function f(x) = [x] is defined as:

f(x) = 1, for all x in [1.2, 1.9)

This is because the greatest integer less than or equal to any number in this interval is 1.

Since f(x) is a constant function in the given interval, its derivative is 0.

for x in [1.2, 1.9)

Hence option 3 is correct