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: Understand the function.
We are given:
\[ f(x) = [2x] \] where \([\,\,]\) denotes the greatest integer function, that is, the greatest integer less than or equal to the given number.

Step 2: Identify the elements whose image is required.
We need the image of the set:
\[ \{-2.3, 2.9\} \] So we will find \(f(-2.3)\) and \(f(2.9)\).

Step 3: Compute \(f(-2.3)\).
First multiply by \(2\):
\[ 2(-2.3) = -4.6 \] Now take the greatest integer less than or equal to \(-4.6\):
\[ [-4.6] = -5 \] Hence,
\[ f(-2.3) = -5 \]

Step 4: Compute \(f(2.9)\).
First multiply by \(2\):
\[ 2(2.9) = 5.8 \] Now take the greatest integer less than or equal to \(5.8\):
\[ [5.8] = 5 \] Hence,
\[ f(2.9) = 5 \]

Step 5: Form the image set.
The image of the set \(\{-2.3, 2.9\}\) under \(f\) is the set of corresponding output values:
\[ \{-5, 5\} \]

Step 6: Check for repetition or simplification.
Since \(-5\) and \(5\) are distinct, both remain in the image set. So the final image is exactly:
\[ \{-5, 5\} \]

Step 7: Match with the options.
The set \(\{-5, 5\}\) is listed in option (2). Therefore, the correct answer is option (2).