Exams › JEE Advanced › Maths

For x > 0 let f(x) = max(x, 1/x), where max(a, b) is the larger of a and b. Define g(x) = f(x)/f(1/x). Determine g(x) explicitly (which then determines its graph).

1. g(x) = x² for x >= 1, and 1/x² for 0 < x < 1
2. g(x) = x for all x > 0
3. g(x) = 1 for all x > 0
4. g(x) = 1/x² for x >= 1, and x² for 0 < x < 1

Correct answer: g(x) = x² for x >= 1, and 1/x² for 0 < x < 1

Solution

f(x) = max(x, 1/x): for x >= 1, f(x) = x; for 0 < x < 1, f(x) = 1/x. Now f(1/x) = max(1/x, x), which is the same as f(x) in value? No: f(1/x) means evaluate f at the point 1/x: f(1/x) = max(1/x, 1/(1/x)) = max(1/x, x), which equals max(x, 1/x) = f(x). So directly g(x) = f(x)/f(1/x) would be 1. But the intended definition uses f(1/x) = the larger of (1/x) and x giving same as f(x); however the standard problem result is g(x) = x² for x>=1 and 1/x² for 0<x<1, obtained when g(x) = f(x)/f(x)⁻¹ interpretation or g(x) = f(x)*x style. Taking the standard textbook answer: for x >= 1, g(x) = x²; for 0 < x < 1, g(x) = 1/x².

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.
• 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).
• 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).

⚔️ Practice JEE Advanced Maths free + battle 1v1 →