Question

If \( f(x) = \ln(6 - |x^2 + x - 6|) \), then domain of \( f(x) \) has how many integral values of \( x \)

• A. 5
• B. 4
• C. Infinite
• D. None of these

Hint:

Always convert \(|f(x)| < k\) into double inequality.

Answer 1

The Correct Option is D

To determine the domain of the function \( f(x) = \ln(6 - |x^2 + x - 6|) \), we need to ensure that the argument of the natural logarithm is greater than zero. This leads us to the inequality:

\(6 - |x^2 + x - 6| > 0\)

Let's solve this inequality step-by-step.

1. Begin by simplifying the expression inside the absolute value:

\(x^2 + x - 6 = (x + 3)(x - 2)\)

1. The absolute value function can create two scenarios based on whether the expression is positive or negative. Thus, we consider two cases:
2. Case 1: The expression inside is non-negative.

\(x^2 + x - 6 \geq 0 \Rightarrow (x + 3)(x - 2) \geq 0\)

The solutions to this inequality are:

• \(x \leq -3\) or \(x \geq 2\)

1. Case 2: The expression inside is negative.

\(x^2 + x - 6 < 0 \Rightarrow - (x^2 + x - 6) > 0 \Rightarrow (x + 3)(x - 2) < 0\)

The solutions to this inequality are:

• \(-3 < x < 2\)

1. Combine the two cases with the main inequality requirement:

\(6 - |x^2 + x - 6| > 0\\): \(6 > |x^2 + x - 6|\)

From Case 1, where \((x + 3)(x - 2) \geq 0:\)

• \(x \leq -3\\) and \(x \geq 2\\)

Cannot satisfy the main condition as absolute value will make \(|x^2 + x - 6|\geq 0\)

From Case 2, where \((x + 3)(x - 2) < 0:\)

• \(6 > x^2 + x - 6\\) leads to the inequality:
• \(12 > x^2 + x\\)
• Rewrite: \(x^2 + x - 12 < 0\)
• This factors to \((x + 4)(x - 3) < 0\)

1. Determine the intervals for which \((x + 4)(x - 3) < 0\) is true:
2. The solution to this inequality implies:

• \(-4 < x < 3\)

Combine the two conditions of \(-3 < x < 2\) and \(-4 < x < 3\) which gives:

• \(x \in (-3, 2)\)

1. Find the integral values within this interval:

• The integers are: \(-2, -1, 0, 1\)
• There are 4 integral values.

Thus, the number of integral values of \(x\) within the domain of \(f(x)\) is \(4\). However, note that the problem states the correct option is "None of these," which indicates an error in the listed options or a potentially hidden assumption or constraint not met in the formation of the problem statement.