Question 995514
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Suppose f(x)=ax^4+bx^2+x+5, where a and b are constants. Find f(4) if f(-4)=3.
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Let us write the fact that  f(-4) = 3.

Simply substitute  x = -4  into the given polynomial.  You will have


f(-4) = {{{a*(-4)^4 + b*(-4)^2 + (-4) + 5}}} = {{{3}}}.


Simplify it,  taking into account that  {{{(-4)^4}}} = {{{4^4}}}  and  {{{(-4)^2}}} = {{{4^2}}}.  You will have 


f(-4) = {{{a*4^4 + b*4^2 + (-4) + 5}}} = {{{3}}}.         (1) 


Now express  f(4).  Simply substitute  x = 4  into the given polynomial.  You will get 


f(4) = {{{a*4^4 + b*4^2 + 4 + 5}}}.                     (2) 


Now compare expressions  (1)  and  (2).  They are almost identical.  The difference is in linear terms  (-4)  in   (1)  and  4  in  (2). 

Hence,  f(4)  is in  8  units greater than  f(-4).  It means that  f(4) = f(-4) + 8 = 3 + 8 = 11.


<U>Answer</U>. &nbsp;f(4) = 11.