Question

The complete range of the function \( f(x) = [x]^2 + 5[x] + 6 \), where \([x]\) is the greatest integer function, is:

• A. \( \mathbb{N} \cup \{0\} \)
• B. \( \mathbb{Z} \)
• C. \( \{ n \mid n = (k-1)k, k \in \mathbb{Z} \} \)
• D. \( \{ n \in \mathbb{N} \mid n = (2+k)(3+k), k \in \{0, 1, 2, \dots\} \} \)

Hint:

When dealing with the greatest integer function in a range problem, replace \([x]\) with an integer variable \(n\) and simplify the algebraic expression to identify the pattern of the output values.

Answer 1

The Correct Option is C

Concept: The greatest integer function \([x]\) outputs an integer \(n\) for any real number \(x\), where \(n \in \mathbb{Z}\). The function can be factored as: \[ f(x) = [x]^2 + 5[x] + 6 = ([x] + 2)([x] + 3) \]

Step 1: Substitute \(n = [x]\) into the factored expression.
Since \(n\) can be any integer, we let \(f(n) = (n+2)(n+3)\).

Step 2: Introduce a substitution to simplify the range.
Let \( k = n + 3 \). As \(n\) ranges over all integers \(\mathbb{Z}\), \(k\) also ranges over all integers \(\mathbb{Z}\). Substituting \(n = k - 3\): \[ f(n) = ((k-3)+2)((k-3)+3) = (k-1)k \]

Step 3: Evaluate the expression for integer values of \(k\).
For \(k \in \mathbb{Z}\), the values of \(k(k-1)\) generate the set of numbers that are products of two consecutive integers (e.g., \( \dots, 6, 2, 0, 0, 2, 6, \dots \)). This matches the set defined in option (C). { n n = (k-1)k, k Z }