Question

Let \( [t] \) denote the greatest integer function and \( [t - m] = [t] - m \) when \( m \in \mathbb{Z} \). If \( k = 2[2x - 1] - 1 \) and \( 3[2x - 2] + 1 = 2[2x - 1] - 1 \), then the range of \( f(x) = [k + 5x] \) is:

• A. $\{7, 8, 9\}$
• B. $\{4, 5, 6\}$
• C. $\{5, 6, 7\}$
• D. $\{6, 7, 8\}$

Hint:

When dealing with the greatest integer function, express variables in terms of integers and analyze the resulting inequalities. Use modular arithmetic to simplify constraints and determine possible integer outputs.

Answer 1

The Correct Option is D

Step 1: Simplify the Given Equation
We start with the equation:
\[ 3[2x - 2] + 1 = 2[2x - 1] - 1 \] Using the property \( [t - m] = [t] - m \) for \( m \in \mathbb{Z} \):
\[ \begin{aligned} [2x - 2] &= [2x] - 2 \\ [2x - 1] &= [2x] - 1 \end{aligned} \] Substitute these into the original equation:
\[ \begin{aligned} 3([2x] - 2) + 1 &= 2([2x] - 1) - 1 \\ 3[2x] - 6 + 1 &= 2[2x] - 2 - 1 \\ 3[2x] - 5 &= 2[2x] - 3 \\ [2x] &= 2 \end{aligned} \] Step 2: Determine the Range of \(x\)
From \( [2x] = 2 \), we have:
\[ 2 \leq 2x < 3 \implies 1 \leq x < 1.5 \] Step 3: Express \(k\) in Terms of \(x\)
Given:
\[ k = 2[2x - 1] - 1 \] Again, using the property:
\[ [2x - 1] = [2x] - 1 \] Substitute \( [2x] = 2 \):
\[ \begin{aligned} k &= 2(2 - 1) - 1 \\ &= 2(1) - 1 \\ &= 1 \end{aligned} \] Step 4: Find the Range of \(f(x) = [k + 5x]\)
With \( k = 1 \), the function becomes:
\[ f(x) = [1 + 5x] \] Evaluate \( f(x) \) over \( x \in [1, 1.5) \):
For \( 1 \leq x < 1.2 \):
\[ 1 + 5x \in [6, 7) \implies [1 + 5x] = 6 \]
For \( 1.2 \leq x < 1.4 \):
\[ 1 + 5x \in [7, 8) \implies [1 + 5x] = 7 \]
For \( 1.4 \leq x < 1.5 \):
\[ 1 + 5x \in [8, 8.5) \implies [1 + 5x] = 8 \]
Thus, the range of \( f(x) \) is \( \{6, 7, 8\} \). Verification
Check specific points:
At \( x = 1.1 \):
\[ [1 + 5(1.1)] = [6.5] = 6 \]
At \( x = 1.3 \):
\[ [1 + 5(1.3)] = [7.5] = 7 \]
At \( x = 1.45 \):
\[ [1 + 5(1.45)] = [8.25] = 8 \]
Conclusion
The range of \( f(x) \) is \( \{6, 7, 8\} \), which corresponds to option \( \boxed{4} \).