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If f(x^2) = x^4–6x^2+5, which of the following must be true?

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If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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New post 24 Mar 2019, 00:31
1
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

58% (02:50) correct 42% (02:03) wrong based on 36 sessions

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If f(x^2) = x^4–6x^2+5, which of the following must be true?

A. f(x^2)=f(1+x^2)
B. f(x^2)=f(2-x^2)
C. f(x^2)=f(3-x^2)
D. f(x^2)=f(5-x^2)
E. f(x^2)=f(6-x^2)
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Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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New post 24 Mar 2019, 07:04
I tried usiing conventional approach,
Substitute the value of \(x^2\) to each value given in option from A to E.
Only E satisfies the given equation
Is their a shorter way to do this.
also is possible to write question int he mathematical formula format.
It would be easier to read
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Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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New post 25 Mar 2019, 02:53
kiran120680 wrote:
If f(x^2) = x^4–6x^2+5, which of the following must be true?

A. f(x^2)=f(1+x^2)
B. f(x^2)=f(2-x^2)
C. f(x^2)=f(3-x^2)
D. f(x^2)=f(5-x^2)
E. f(x^2)=f(6-x^2)


You can factorize the above eqn to (x^2-5)(x^2-1)..then it is easy
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Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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New post 07 Apr 2019, 10:46
Can someone please explain it in detail?

Posted from my mobile device
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Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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New post 07 Apr 2019, 12:12
1
arorni wrote:
Can someone please explain it in detail?

Posted from my mobile device


Let me try
\(f(x^2) = x^4–6x^2+5\), which of the following must be true?

Lets factor the given equation first
\(x^4–6x^2+5\) = \((x^2-5)(x^2-1)\)

A.\(f(x^2)=f(1+x^2)\)
B.\(f(x^2)=f(2-x^2)\)
C. \(f(x^2)=f(3-x^2)\)
D. \(f(x^2)=f(5-x^2)\)
E. \(f(x^2)=f(6-x^2)\)

Now for each options put the given values of f(n) instead of f(x^2) in the factor
\((x^2-5)(x^2-1)\)
Such that for example if you put
\((1+x^2)\) instead of \(x^2\) in the function value \((x^2-5)(x^2-1)\) . It will become
=\((1+x^2-5)(1+x^2-1)\)=\((+x^2-4)(x^2)\). This is not equal to original expression.

You will find only E fits in substitute\((6-x^2)\) in place of\(x^2\)

\((x^2-5)(x^2-1) = (6-x^2-5)(6-x^2 -1)= (x^2-1)(x^2-5)\).
This is equal to original expression. Hence (E)
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If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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New post 07 Apr 2019, 13:35
1
kiran120680 wrote:
If f(x^2) = x^4–6x^2+5, which of the following must be true?

A. f(x^2)=f(1+x^2)
B. f(x^2)=f(2-x^2)
C. f(x^2)=f(3-x^2)
D. f(x^2)=f(5-x^2)
E. f(x^2)=f(6-x^2)

arorni wrote:
Can someone please explain it in detail?

You can reduce the difficulty of the question once you realize that the function is quadratic and not of 4th degree.

Setting \(z = x^2\), one can see that the function \(f\) itself is \(f(z) = z^2 -6z +5\), and the question is: "For which \(\beta\) is \(f(\beta -z) = f(z)\) true for all \(z \geq 0\)?" (modify it for answer choice A). In particular, the relation is true for \(z = 0\) which boils down to \(f(\beta) = f(0)\) or \(\beta^2 -6\beta +5 = 5\), and therefore \(\beta = 6\).

In practice, you simply calculate \(f(1), f(2), f(3), f(5)\), and \(f(6)\) and look which one returns the value 5.
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If f(x^2) = x^4–6x^2+5, which of the following must be true?   [#permalink] 07 Apr 2019, 13:35
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If f(x^2) = x^4–6x^2+5, which of the following must be true?

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