If f(x) = frac2^x2^x + sqrt2$f(x) = \frac{2^x}{2^x + \sqrt{2}}$, x in mathbbR$x \in \mathbb{R}$, then sum_k=1^81 fleft(frack82right)$\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to: (1) 41 (2) frac812$\frac{81}{2}$ (3) 82 (4) 81sqrt2$81\sqrt{2}$

### Core Logic f(x) = frac4x + 36x - 4$$f(x) = \frac{4x + 3}{6x - 4}$$ Compute g(x) = f(f(x))$g(x) = f(f(x))$: g(x) = frac4left(frac4x + 36x - 4right) + 36left(frac4x + 36x - 4right) - 4 = frac16x + 12 + 18x - 1224x + 18 - 24x + 16 = frac34x34 = x$$g(x) = \frac{4\left(\frac{4x + 3}{6x - 4}\right) + 3}{6\left(\frac{4x + 3}{6x - 4}\right) - 4} = \frac{16x + 12 + 18x - 12}{24x + 18 - 24x + 16} = \frac{34x}{34} = x$$ ### Step 1: Composition Evaluation Since g(x) = x$g(x) = x$, g$g$ is the identity function. (gogog)(4) = g(g(g(4))) = 4$$(gogog)(4) = g(g(g(4))) = 4$$ ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Relations and Functions

### Core Logic S$S$ must be reflexive, symmetric, and transitive, containing (1,2)$(1,2)$, (2,3)$(2,3)$, and (1,4)$(1,4)$. Symmetric property forces (2,1), (3,2), (4,1) in S$(2,1), (3,2), (4,1) \in S$. Transitive property: (1,2)$(1,2)$ and (2,3) implies (1,3) in S$(2,3) \implies (1,3) \in S$. Symmetric implies (3,1) in S$\implies (3,1) \in S$. (4,1)$(4,1)$ and (1,2) implies (4,2) in S$(1,2) \implies (4,2) \in S$. Symmetric implies (2,4) in S$\implies (2,4) \in S$. (4,1)$(4,1)$ and (1,3) implies (4,3) in S$(1,3) \implies (4,3) \in S$. Symmetric implies (3,4) in S$\implies (3,4) \in S$. ### Step 1: Universal Relation Since 1 is related to 2, 3, 4 and the relation is an equivalence relation (which creates partitions), all elements 1, 2, 3, and 4 must fall into the same single equivalence class. Thus, S$S$ must contain all possible ordered pairs in A times A$A \times A$. ### Step 2: Final Count Number of elements in A times A = 4 times 4 = 16$A \times A = 4 \times 4 = 16$. Minimum value of n$n$ is 16$16$. ### Pattern Recognition If a relation connects all elements in a set to each other through a chain, its equivalence closure is the universal relation A times A$A \times A$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Relations and Functions