Mathematical Function and Inverse

Function Understanding \n - Function Definition: \n - The function is defined as ( f(x) = \sqrt{\sqrt{2x + 3}} ). \n - Components of the Function: \n - The outer square root applies to the inner expression, which is itself a square root applied to (2x + 3). \n \n# Finding the Inverse Function \n - Notation: The inverse function is denoted as ( f^{-1}(a) ). \n - Objective: Find ( f^{-1}(5) ). \n - Method: To find the inverse, we want to express ( x ) in terms of ( y ) (where ( y = f(x) )). \n \n# Steps to Find Inverse \n 1. Set the function equal to y: \n - Let ( y = \sqrt{\sqrt{2x + 3}} ) \n 2. Square both sides to eliminate the outer square root: \n - ( y^2 = \sqrt{2x + 3} ) \n 3. Square both sides again to eliminate the inner square root: \n - ( (y^2)^2 = 2x + 3 ) \n - This simplifies to: ( y^4 = 2x + 3 ) \n 4. Rearranging for x: \n - Subtract 3 from both sides: ( y^4 - 3 = 2x ) \n - Divide everything by 2: ( x = \frac{y^4 - 3}{2} ) \n 5. Express as inverse function: \n - Hence, ( f^{-1}(y) = \frac{y^4 - 3}{2} ) \n \n# Finding Specific Value \n - Substituting 5 into the inverse function: \n - To find ( f^{-1}(5) ): \n - ( f^{-1}(5) = \frac{5^4 - 3}{2} ) \n - Calculate Step by Step: \n 1. Calculate ( 5^4:) \n - ( 5^4 = 625 ) \n 2. Substitute into the equation: ( \frac{625 - 3}{2} ) \n 3. Simplifying: ( \frac{622}{2} = 311 ) \n \n# Conclusion \n - Final Result: ( f^{-1}(5) = 311 ) \n - Correctness: The initial response to the query was incorrect, and the solution should reflect the true result where ( f^{-1}(5) = 311. Note: Review and ensure understanding of function inverses and operations on roots.