AP EAMCET202315 May 2023Evening ShiftMathematicsFunctionsActual
If a real valued function f is defined by f(x)= a x+ a^2-x^2 b x , then f is
Options
- Aonly one-one
- Bonly onto
- Cboth one-one and onto
- Dneither one-one nor onto
Correct answer
C. both one-one and onto
Step-by-step solution
f(x)= a x+ x^2-a^2 b x Then f^ (x)= -a^2 b x^2 x^2-a^2 if b>0 f^ (x) 0 f(x) is increasing f(x) is one-one. Now, let y= a x+ a^2-x^2 b x aligned & (b x y-a x)^2=a^2-x^2 & x^2(b y-a)^2+x^2=a^2 & x^2= a^2 1+(b y-a)^2 & x= a 1+(b y-a)^2 [ 1+(b y-a)^2>0 ] & y R aligned So, f ( x ) is onto also.