Question

If \( A \subseteq \mathbb{Z} \) and the function \( f: A \to \mathbb{R} \) is defined by \[ f(x) = \frac{1}{\sqrt{64 - (0.5)^{24+x- x^2} }} \] then the sum of all absolute values of elements of \( A \) is:

• A. \( 36 \)
• B. \( 5 \)
• C. \( 25 \)
• D. \( 11 \)

Hint:

When dealing with square root functions, always ensure the expression inside the square root is positive. This helps determine the domain of the function.

Answer 1

The Correct Option is C

To determine the sum of all absolute values of elements of \( A \), we first need to identify the domain \( A \) of the function \( f(x) \). The function is defined as: \[ f(x) = \frac{1}{\sqrt{64 - (0.5)^{24 + x - x^2}}} \] For the function to be real-valued, the expression inside the square root must be positive: \[ 64 - (0.5)^{24 + x - x^2}>0 \] This simplifies to: \[ (0.5)^{24 + x - x^2}<64 \] Since \( 0.5 = 2^{-1} \) and \( 64 = 2^6 \), we can rewrite the inequality as: \[ 2^{-(24 + x - x^2)}<2^6 \] Taking the logarithm base 2 of both sides (and remembering that the inequality direction remains the same since the logarithm is increasing): \[ -(24 + x - x^2)<6 \] Multiply both sides by \(-1\) (which reverses the inequality): \[ 24 + x - x^2>-6 \] Rearrange the inequality: \[ -x^2 + x + 30>0 \] Multiply both sides by \(-1\) (again reversing the inequality): \[ x^2 - x - 30<0 \] Now, solve the quadratic inequality \( x^2 - x - 30<0 \). First, find the roots of the equation \( x^2 - x - 30 = 0 \): \[ x = \frac{1 \pm \sqrt{1 + 120}}{2} = \frac{1 \pm \sqrt{121}}{2} = \frac{1 \pm 11}{2} \] So, the roots are: \[ x = \frac{12}{2} = 6 \quad \text{and} \quad x = \frac{-10}{2} = -5 \] The quadratic \( x^2 - x - 30 \) is a parabola opening upwards. Therefore, it is negative between its two roots. Thus, the inequality \( x^2 - x - 30<0 \) holds for: \[ -5<x<6 \] Since \( x \) is an integer (\( x \in \mathbb{Z} \)), the possible values of \( x \) are: \[ x \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \] The set \( A \) is: \[ A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \] Now, compute the sum of the absolute values of all elements in \( A \): \[ |-4| + |-3| + |-2| + |-1| + |0| + |1| + |2| + |3| + |4| + |5| = 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 25 \] Therefore, the sum of all absolute values of elements of \( A \) is: \[ \boxed{25} \]