Permutations

We are given the function:

f(x)%20%3D%20%5Cfrac%7Ba%5Ex%20%2B%20a%5E%7B-x%7D%7D%7B2%7D.

Now, let's calculate

f(x%2By)

and

f(x-y)

f(x%2By)%20%3D%20%5Cfrac%7Ba%5E%7Bx%2By%7D%20%2B%20a%5E%7B-(x%2By)%7D%7D%7B2%7D%2C%20%5Cquad%20f(x-y)%20%3D%20%5Cfrac%7Ba%5E%7Bx-y%7D%20%2B%20a%5E%7B-(x-y)%7D%7D%7B2%7D.

By adding these two expressions together:

f(x%2By)%20%2B%20f(x-y)%20%3D%20%5Cfrac%7Ba%5E%7Bx%2By%7D%20%2B%20a%5E%7B-(x%2By)%7D%20%2B%20a%5E%7Bx-y%7D%20%2B%20a%5E%7B-(x-y)%7D%7D%7B2%7D.

Next, simplify this using the properties of exponents:

f(x%2By)%20%2B%20f(x-y)%20%3D%20%5Cfrac%7Ba%5Ex(a%5Ey%20%2B%20a%5E%7B-y%7D)%20%2B%20a%5E%7B-x%7D(a%5Ey%20%2B%20a%5E%7B-y%7D)%7D%7B2%7D.

Factor the expression:

f(x%2By)%20%2B%20f(x-y)%20%3D%20%5Cfrac%7B(a%5Ex%20%2B%20a%5E%7B-x%7D)(a%5Ey%20%2B%20a%5E%7B-y%7D)%7D%7B2%7D.

Now, recognizing that

f(x)%20%3D%20%5Cfrac%7Ba%5Ex%20%2B%20a%5E%7B-x%7D%7D%7B2%7D

and

f(y)%20%3D%20%5Cfrac%7Ba%5Ey%20%2B%20a%5E%7B-y%7D%7D%7B2%7D

, we substitute these back in:

f(x%2By)%20%2B%20f(x-y)%20%3D%202f(x)f(y).

Final Answer:

%5Cboxed%7B2f(x)f(y)%7D

Chapter Questions
3 MCQs