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f: RvecR ,f(x^2=x+3)+2f(x^2-3x+5)=6x^2-1...

`f: RvecR ,f(x^2=x+3)+2f(x^2-3x+5)=6x^2-10 x+17AAx in R ,` then find the function `f(x)dot`

Text Solution

Verified by Experts

The correct Answer is:
`f(x)=2x-3`
Obviously, `f` is a linear polynomial.
Let `f(x)=ax+b.` Hence,
`f(x^(2)+x+3)+2f(x^(2)-3x+5)=6x^(2)-10x+17`
or ` [a(x^(2)+x+3)+b]+2[a(x^(2)-3x+5)+b]=6x^(2)-10x+17`
or `a+2a=6 " (1) " `
and `a-6a= -10 " (2)" `
or `a=2 " " ` (Comparing coeff. of `x^(2)` and coeff. of `x` both sides)
Again, `3a+b+10a+2b=17`
or `6+b+20+2b=17`
` :. b= -3`
or `f(x)=2x-3`
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