A and B are respectively the domain and range of a real valued function f(x) = [Sin^-1((10x+6)/3)]. then the number of integers in A union B is ([y] represents the greatest integer less than or equal to y)

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q2 – AP-EAMCET 2026

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Home Maths Relations and FunctionsQuestion #359

A and B are respectively the domain and range of a real valued function

f (x) = [{Sin}^{- 1} (\frac{10 x + 6}{3})]

. then the number of integers in

A \cup B

is ([y] represents the greatest integer less than or equal to y)

Clue 1 of 3: Domain and Range Definition

Understand that for a function $f (x) = g (h (x))$ , the domain is determined by the conditions for $h (x)$ to be defined, and the range is determined by the possible values of $g$ after $h (x)$ is evaluated.

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Step 1: Determine the Domain (A) of the function✦ Active

The function is $f (x) = [{Sin}^{- 1} (\frac{10 x + 6}{3})]$ . For the inverse sine function, ${Sin}^{- 1} (u)$ , to be defined, its argument $u$ must be in the interval $[- 1, 1]$ . Here, $u = \frac{10 x + 6}{3}$ .

- 1 \leq \frac{10 x + 6}{3} \leq 1

Multiply all parts by 3:

- 3 \leq 10 x + 6 \leq 3

Subtract 6 from all parts:

- 3 - 6 \leq 10 x \leq 3 - 6 - 9 \leq 10 x \leq - 3

Divide all parts by 10:

- \frac{9}{10} \leq x \leq - \frac{3}{10} - 0.9 \leq x \leq - 0.3

Thus, the domain $A = [- 0.9, - 0.3]$ . There are no integers in this interval.

Step 2: Determine the Range (B) of the function○ Expand

First, consider the range of ${Sin}^{- 1} (\frac{10 x + 6}{3})$ . Since the argument $\frac{10 x + 6}{3}$ spans the entire domain of ${Sin}^{- 1}$ (i.e., $[- 1, 1]$ ), the range of ${Sin}^{- 1} (\frac{10 x + 6}{3})$ is its principal value range, which is $[- \frac{π}{2}, \frac{π}{2}]$ . Numerically, $\frac{π}{2} \approx 1.57$ .

- 1.57 \leq {Sin}^{- 1} (\frac{10 x + 6}{3}) \leq 1.57

Next, apply the greatest integer function $[y]$ (also known as the floor function $⌊ y ⌋$ ) to this interval. The greatest integer function returns the largest integer less than or equal to $y$ . For $y \in [- 1.57, 1.57]$ , the possible integer values are:

{\begin{cases} - 2 & if y \in [- 1.57, - 1) \\ - 1 & if y \in [- 1, 0) \\ 0 & if y \in [0, 1) \\ 1 & if y \in [1, 1.57] \end{cases}

Therefore, the set of all possible integer values for $f (x)$ , which is the range $B$ , is ${- 2, - 1, 0, 1}$ .

💡 Teacher's Secret Hint

Remember that the greatest integer function always rounds down to the nearest integer. For example, $[- 1.57] = - 2$ and $[1.57] = 1$ .

Step 3: Find the number of integers in

A \cup B

○ Expand

We have the domain $A = [- 0.9, - 0.3]$ and the range $B = {- 2, - 1, 0, 1}$ .

The union of these two sets is $A \cup B = [- 0.9, - 0.3] \cup {- 2, - 1, 0, 1}$ .

To find the number of integers in $A \cup B$ , we list the integers from both sets. Set $A$ contains no integers. Set $B$ contains the integers ${- 2, - 1, 0, 1}$ .

The integers in $A \cup B$ are thus ${- 2, - 1, 0, 1}$ .

There are 4 distinct integers in $A \cup B$ .

💡 Teacher's Secret Hint

Carefully identify integers. Integers are whole numbers, positive, negative, or zero. Ensure you don't double-count any elements if there were overlaps.

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Maths Question 2 – AP-EAMCET 2026

Clue 1 of 3: Domain and Range Definition

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