Question
If the function f : R → R is defined by f x = x x − sin x , then which of the following statements is TRUE?
JEE Advanced
Mathematics
Functions
2020
JEE Advanced 2020 (Paper 1)
JEE Advanced Mathematics Question (2020) — Solution
Options
- A. f is one-one, but NOT onto
- B. f is onto, but NOT one-one
- C. f is BOTH one-one and onto
- D. f is NEITHER one-one NOR onto
Answer
C. f is BOTH one-one and onto
Step-by-step solution
f x = x x − sin x is odd function ∵ f − x = − f x Now, f x = x 2 − x sin x ∀ x ≥ 0 ⇒ f ' x = 2 x − x cos x − sin x ⇒ f ' x = x − sin x + x 1 − cos x > 0 ∴ Graph of y = f x will be From graph, it's one-one and onto both.
Related: Mathematics — Functions · All PYQ Banks