Question

If \(f(x) = [2x]\), where \([x]\) denotes the greatest integer function in \(x\), then the image of \(\{-2.3, 2.9\}\) is

• A. \(\{-5, 3\}\)
• B. \(\{-5, 5\}\)
• C. \(\{-4, 5\}\)
• D. \(\{-3, 2\}\)
• E. \(\{-4, 6\}\)

Hint:

In the greatest integer function, \([x]\) means the greatest integer less than or equal to \(x\). Be extra careful for negative numbers: for example, \([-4.6] = -5\), not \(-4\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the "image" of a set under a given function. The image is the set of all output values obtained by applying the function to each element of the input set. The function is \(f(x) = [2x]\), which uses the greatest integer function (or floor function). The greatest integer function \([y]\) yields the largest integer that is less than or equal to \(y\).
Step 2: Detailed Explanation:
The input set is \{-2.3, 2.9\}. We need to compute the function's value for each element in this set.
1. Compute f(-2.3):
Substitute x = -2.3 into the function:
\[ f(-2.3) = [2 \times (-2.3)] = [-4.6] \] The greatest integer less than or equal to -4.6 is -5. (On a number line, -4.6 lies between -5 and -4. The integer to its immediate left is -5).
So, \(f(-2.3) = -5\).
2. Compute f(2.9):
Substitute x = 2.9 into the function:
\[ f(2.9) = [2 \times 2.9] = [5.8] \] The greatest integer less than or equal to 5.8 is 5. (On a number line, 5.8 lies between 5 and 6. The integer to its immediate left is 5).
So, \(f(2.9) = 5\).
Step 3: Final Answer:
The set of all output values (the image) is composed of the results we found.
Image = \{-5, 5\}.
Therefore, option (B) is the correct answer.