Question

If a + α = 1, b + β = 2 and a f(x) + α f(1/x) = b x + β / x, then [f(x) + f(1/x)] / [x + 1/x] is ________

Hint:

In functional equations involving $f(x)$ and $f(1/x)$, replacing $x$ with $1/x$ usually creates a system of linear equations that can be solved by addition or elimination.

Answer 1

Correct Answer: 2

Step 1: Given $a f(x) + \alpha f(1/x) = bx + \beta/x$ (i).
Step 2: Replace $x$ by $1/x$: $a f(1/x) + \alpha f(x) = b/x + \beta x$ (ii).
Step 3: Add (i) and (ii): $(a+\alpha)f(x) + (a+\alpha)f(1/x) = (b+\beta)x + (b+\beta)/x$.
Step 4: Substitute $a+\alpha=1$ and $b+\beta=2$: $1[f(x) + f(1/x)] = 2[x + 1/x]$.
Step 5: Result is 2.