If a + α = 1, b + β = 2 and af(x) + α f( 1/x ) = bx + β/x,x ≠ 0, then the value of the expression (f(x) + f( 1/x ))/(x + 1/x) is __________.

Solution

Maths sir · 4 steps

Swap x with 1/x and add the two equations. The sum f(x) + f(1/x) then drops right out.

1. Replace x by 1/x. The given equation is af(x) + α f(1/x) = bx + β/x. Putting 1/x in place of x gives a second equation: af(1/x) + α f(x) = b/x + β x.
2. Add both equations. On the left, f(x) and f(1/x) each get the coefficient a + α. On the right, x and 1/x each get the coefficient b + β.

   (a+α)[f(x) + f(1/x)] = (b+β)(x + 1/x)

3. Use the given sums. Put a + α = 1 and b + β = 2. This gives f(x) + f(1/x) = 2(x + 1/x).
4. So. Dividing by x + 1/x, the expression equals (b+β)/(a+α) = 2/1 = 2.

The common slip: 0.5. Turning the ratio upside down as (a+α)/(b+β) gives 0.5, not 2. The sum b + β multiplies x + 1/x, so it belongs on top.

The idea to keep. When a functional equation pairs f(x) with f(1/x), write it again with x → 1/x. Adding the two equations gives the sum, and subtracting them gives the difference.

Answer from the NTA final answer key. Solution by PrepWiser.