Solve this: Q 10 Let f : R →R, then f (x) = 2x + | cos x | - Maths - Relations and Functions

Question

Solve this:
Q 10 Let f : R → R, then f (x) = 2x + | cos x | is
(A) one-one and into
(B) one-one and onto
(C) many-one and into
(D) many-one and onto

Aarushi Mishra · Accepted Answer

Note:d xdx=xx=±1, x≠0fx=2x+cos xNote:As x→∞, fx→∞ and as x→-∞, fx→-∞Since fx is continuous, therefore fx will take all values between -∞ and∞Range fx=co-domain=ℝf'x=d2xdx+d dxcos xf'x=2+cos xcos xd cos xdxf'x=2+cos xcos x-sin x,          cos x≠0Since cos xcos x=±cos xcos x=±1f'x=2±sin x,          cos x≠0f'x=2+sin x or f'x=2- sin x-1≤sin x≤12-1≤2+sin x≤2+1-1≤sin x≤11≥-sin x≥-11+2≥2-sin x≥2-13≥2-sin x≥1Hence f'x>0Function is increasing so it will be one-onefx is one-one and onto

Solve this: Q.10. Let f : R → R, then f (x) = 2x + | cos x | is (A) one-one and into (B) one-one and onto (C) many-one and into (D) many-one and onto

Solve this:
Q.10. Let f : R $\to$ R, then f (x) = 2x + | cos x | is
(A) one-one and into
(B) one-one and onto
(C) many-one and into
(D) many-one and onto