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How many common points have y=|x| and y=|x^2-4|?

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How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post Updated on: 10 Apr 2019, 14:02
4
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  85% (hard)

Question Stats:

38% (01:52) correct 63% (01:45) wrong based on 88 sessions

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How many common points have \(y=|x|\) and \(y=|x^2 - 4|\)?

A. 0
B. 1
C. 2
D. 4
E. Infinitely many

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MEMENTO AUDERE SEMPER

Originally posted by lucajava on 10 Apr 2019, 07:12.
Last edited by lucajava on 10 Apr 2019, 14:02, edited 2 times in total.
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Re: How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post 10 Apr 2019, 12:19
5
Algebraic way (solving the moduli case by case) is not difficult, but if you sketch the graph of the functions you will save precious time.
You just need to figure out that \(y = |x|\) bisects the first and third quadrant for \(x ≥ 0\) and second and fourth otherwise, meanwhile \(y = |x^2 - 4|\) is an upward-opening parabola for \(x ≤ -2\) and \(x ≥ 2\), and a downward-opening one for \(-2 < x < 2\). Count the number of intersections: 4. Et voilà, les jeux sont faits.
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MEMENTO AUDERE SEMPER
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Re: How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post 10 Apr 2019, 08:33
lucajava wrote:
How many common points do \(y=|x|\) and \(y=|x^2 - 4|\) have?

A. 0
B. 1
C. 2
D. 4
E. Infinitely many




can someone explain this question please?

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Re: How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post 10 Apr 2019, 11:13
1
Given y= |x|
thus y= x when x>0
y = -x when x<0
Similarly y=x^2−4 when\(|x^2−4| >0\) or x>2 or x<-2
y = -(x^2−4) when \(|x^2−4| <0\) or when 2>x>-2
I used the above 4 equation and solved values of x
4 such roots satisfies
using\(b^2 -4ac\)
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Re: How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post 09 Oct 2019, 08:16
lucajava wrote:
How many common points have \(y=|x|\) and \(y=|x^2 - 4|\)?

A. 0
B. 1
C. 2
D. 4
E. Infinitely many


\(intersection(common.points):|x|=|x^2-4|\)
\(y=|x|…y=0:x≥0…y=-x:x<0\)
\(y=|x^2 - 4|…y=x^2 - 4:(x+2)(x-2)≥0=…x>2,x<-2…\)
\(y=|x^2 - 4|…y=-x^2+4:(x+2)(x-2)<0=…-2<x<2\)

\(range: ----(-2)----0----(2)----\)
\((x>2):x=x^2-4…x^2-x-4=0…b^2-4ac=1-4(-4)=17=2.points\)
\((0<x<2):x=-x^2+4…x^2+x-4=0…1-4(-4)=17=2.points\)
\((-2<x<0):-x=-x^2+4…x^2-x-4=0…already.tested\)
\((x<-2):-x=x^2-4…x^2+x-4=0…already.tested\)

\(total.points=2+2=4\)

Answer (D)
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Re: How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post 09 Oct 2019, 08:57
4
Let's solve this question with the help of graphs

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Re: How many common points have y=|x| and y=|x^2-4|?  [#permalink]

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New post 09 Oct 2019, 12:34
satya2029 wrote:
Let's solve this question with the help of graphs

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Followed the same approach. Graphing the question makes it so much easier to solve.
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Re: How many common points have y=|x| and y=|x^2-4|?   [#permalink] 09 Oct 2019, 12:34
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