pls answer this f:R- R, f(x)=(ax^2 +6x -8)/(a+6x-8x^2) is onto then range of a is - Maths - Relations and Functions

Question

pls answer this
f:R->R, f(x)=(ax^2 +6x -8)/(a+6x-8x^2) is onto then range of a
is?

Ashish Shukla · Accepted Answer

Dear Student,
y=a2+6x-8a+6x-8x2y(a+6x-8x2)=a2+6x-8(a+8y)x2+6(1-y)x-(8+ay)=0since, x is real
D≥0i e
===>b2-4ac≥036(1-y)2+4(a+8y)(8+ay)≥09(1-2y+y2)+(8a+a2y+64a+8ay2)≥0y2(9+8a)+y(a2+46)+(9+8a)≥0
(i)i will hold for each y∈R if 9+8a>0 and b2-4ac , i
e
 (46+a2)2-4(9+8a)2≤0Therefore,a>-98 and 46+a2-2(9+8a)46+a2+2(9+8a)≤0    ( use A2-B2=(A+B)(A-B))a>-98 and a2-16a+28a2+16a+64)≤0  a>-98 and (a-2)(a-14)(a+8)2≤0a>-98 and (a-2)(a-14)≤0
(since (a+8)2≥0, as square can never be negative)a>-98 and 2≤a≤14Therefore,2≤a≤14
Hence f(x) will be onto if 2≤a≤14Therefore,Required range of a will be 2,14
Hope this helps you clear your doubt
Regards

pls answer this f:R->R, f(x)=(ax^2 +6x -8)/(a+6x-8x^2) is onto then range of a is?

pls answer this
f:R->R, f(x)=(ax^2 +6x -8)/(a+6x-8x^2) is onto then range of a is?