Symbolic Polynomial
Given that
f ( x 2 + 3 ) f ( x 2 − 2 ) = x 4 + 6 x 2 + 3 and = x 4 − 4 x 2 − 2
find f ( 1 ) ?
The answer is -5.
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1 solution
What do you mean by x < − 2 ?
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I am not quite sure what you mean. I wrote that we know nothing about the function ( f ( x ) ) if x < − 2 , and that is correct.
We do know that f ( − 2 ) = f ( 0 2 − 2 ) = 0 4 − 4 ⋅ 0 2 − 2 = − 2 . But we know nothing about, say, f ( − 3 ) ; here the function could have any imaginable value.
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I read my comment and changed it, didn't know what I did to make that sentence, but this was my question and your reply answered it ;)
Let x be such that x 2 = 3 . (Such a real number exists.)
Then f ( 1 ) = f ( 3 − 2 ) = f ( x 2 − 2 ) = x 4 − 4 x 2 − 2 = 3 2 − 4 ⋅ 3 − 2 = 9 − 1 2 − 2 = − 5 .
If x is limited to real numbers, then the first equation is useless, since there is no x such that x 2 + 3 = 1 . However, if we allow complex numbers, let x 2 = − 2 . Then f ( 1 ) = f ( x 2 + 3 ) = ( − 2 ) 2 + 6 ⋅ ( − 2 ) + 3 = 4 − 1 2 + 3 = − 5 . Thus this problem may be solved with real numbers as well as complex numbers.
The given conditions may be written as f ( x 2 + 3 ) = ( x 2 + 3 ) 2 − 6 ; f ( x 2 − 2 ) = ( x 2 − 2 ) 2 − 6 . This suggests that in general, f ( x ) = x 2 − 6 . However, to prove that this must necessarily be true for all x requires more work! In fact, we know nothing about the behavior of the function if x < − 2 . This generalization is therefore not justified; it is safer, as I did above, only to draw conclusions about the specific value f ( 1 ) .