Exams › JEE Advanced › Maths

A polynomial function satisfies f(x)*f(1/x) = f(x) + f(1/x) for all x != 0, and f(3) = -26. Find f(4).

1. -15
2. -63
3. -47
4. -255

Correct answer: -63

Solution

The equation f(x)f(1/x) = f(x)+f(1/x) rearranges to (f(x)-1)(f(1/x)-1) = 1. For a polynomial, f(x)-1 must be a monomial: f(x)-1 = +/-xⁿ. Taking f(x) = 1 - xⁿ and using f(3) = 1-27 = -26 gives n=3. Thus f(4) = 1-64 = -63.

Related JEE Advanced Maths questions

• Let f be a polynomial function defined on non-negative real numbers satisfying f(x)*f(y) + 2 = f(x) + f(y) + f(x*y) for all x, y >= 0. Given that f is a one-one function with f(0) = 1 and f'(1) = 2, find the value of f(5).
• Let S = {a in R: |2a - 1| = 3[a] + 2{a}}, where [.] denotes the greatest integer function and {.} denotes the fractional part. Find the value of 4 * (sum of all a in S).
• The range of the function f(x) = (x - 1) / (p - x² + 1) does not contain any value in the interval [-1, -1/3]. Find the true set of values of p.
• Let f: R -> R and g: R -> R be defined by f(x) = ln(x² + 1) - e^(-x) + 1 and g(x) = (1 - 2e^(2x)) / e^x. For which of the following ranges of alpha does the inequality f(g(((alpha-1)²)/3)) > f(g(alpha - 5/3)) hold?
• Let f denote the greatest integer (floor) function and g the modulus (absolute value) function. Evaluate (g o f)(-5/3) - (f o g)(-5/3).
• Determine the range of each function: (i) f(x) = 2/x; (ii) f(x) = 1/(x² - x + 1); (iii) f(x) = (x² - x + 1)/(x² + x + 1).

⚔️ Practice JEE Advanced Maths free + battle 1v1 →