Types Of Functions

Since is an even function, it satisfies . For , being an odd function, it satisfies . Given , substituting into the function , we have: This implies . To express and in terms of their coefficients, we use: Given , it follows that: Therefore, equals , matching the correct answer option.

The sequence given is a series of products of consecutive odd numbers taken three at a time. The general term for the sequence can be expressed as , where is the term number starting from 1. To find , consider the sum of the first two terms of the series: Given that this sum is expressed by the formula for , we have: Thus, , which confirms the correct answer as option (4).

Given the function:

Step 1: Checking the Denominator The denominator of the function is: To analyze this expression, consider different cases for :

Case 1: For , we have , so: Thus, the denominator becomes , making the function undefined.

Case 2: For , we have , so: Again, the denominator becomes , making the function undefined.

Step 2: Conclusion Since the function is undefined for all , it does not map any values from set to set . This implies that and must be disjoint, meaning:

Final Answer:

The given function satisfies the condition . Given this property, to solve for , we first determine the values of and by applying quadratic properties and symmetry. After computing these values and their sum, we find the square root of their sum to be .