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Intern  Joined: 27 Sep 2010
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The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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5
29 00:00

Difficulty:   75% (hard)

Question Stats: 59% (02:20) correct 41% (02:33) wrong based on 356 sessions

### HideShow timer Statistics Attachment: pic.JPG [ 4.77 KiB | Viewed 21098 times ]
The figure above shows the graph of a function f, defined by f(x) = |2x| + 4 for all numbers x. For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ?

A. g(x) = x - 2
B. g(x) = x + 3
C. g(x) = 2x - 2
D. g(x) = 2x + 3
E. g(x) = 3x - 2
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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13
1
bgpower wrote:
Hi guys,

I am actually quite surprised by my capability to even come close to solving this 75% difficulty questions. The only thing I don't understand here is why we're looking for a solution in the 1st quadrant. For example, bagrettin refers to this in his solution.

Answer choices (C) and (D) offer solutions of $$-\frac{1}{2}$$ and $$-\frac{7}{4}$$ in the x<0 or 2nd quadrant area. Why do we not except these answers here?

Thank you!

We are looking for intersection in the first quadrant because all lines in the given five options have positive slope. A line with positive slope definitely passes through the first and third quadrant. So we are ignoring second quadrant for the time being.

If you understand how lines are drawn on xy axis, you can do this question in 10 seconds.

The graph of f(x) is shown. The line in first quadrant has slope 2.

Let's plot all lines on the xy axis.

Attachment: Lines and Slopes.jpg [ 1.14 MiB | Viewed 19445 times ]

We know y = x is a line making 45 degree angle with y axis and passing through the centre. Point at it in the given figure.
y = x - 2 will be two steps below it passing through (0, -2)
y = x + 3 will be two steps above it passing through (0, 3)
They both have slope 1 which is less than the slope of f(x). They will not intersect f(x).

Now hold on to y = 2x, a line passing through (0, 0) but with a steeper slope. The slope is the same as that of f(x) so it is parallel to f(x).
Both y = 2x - 2 and y = 2x + 3 (blue lines) will have slopes parallel to f(x) and will not intersect it.

The only remaining line is 3x - 2 (green line) which is steeper than f(x) and will intersect it.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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7
3
Obviously, this is 3x-2 because it has the slope of 3, so it increases faster than f(x) with its slope of 2.
More formally, the right part of f(x) is given by the equation 2x+4. If graphs of 2 functions have the point of intersection then the following equation should have a solution with x>=0:
2x+4=3x-2
x=6
y=2*6+4=16

So, these lines reaaly intersect each other.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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I did not get this.

y = -2x + 4

y = 2x + 4

C. g(x) = 2x - 2 and D. g(x) = 2x + 3 are || so will not intersect

Let us take g(x) = x - 2

x-2 = -2x + 4 and x - 2 = 2x + 4

=> x = 2 and x = -6

=> y = 0, and y = -8

So (1) also intersects, but the OA given is E.

Could someone please explain where I'm going wrong ?
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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I think, I found the solution - ...

if we take the slope as tangent(x) then we notice that 60<x<90, as tan(x)=2 and tan(x)>Sqrt(3)

Now if we take answer choice (E) we get tan(x)>2 --> angle x>angle x

the graph g(x)=3x-2 has to cross the graph f at any given time, because of the angle property (x) and disposition of adjacent (rise in x-coordinate) and opposite (rise in y-coordinate) sides.

all other answer choices could be parallel or placed under the graph {fixed y-coordinate in f(x)} too, they depend on the value of x. We can check simply by plugging in x=0 and x=1 into answer choices A-D.

thanks
Attachments pic.JPG [ 9.86 KiB | Viewed 20990 times ]

Originally posted by zaur2010 on 28 Mar 2011, 20:41.
Last edited by zaur2010 on 28 Mar 2011, 20:51, edited 1 time in total.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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I didn't get the right answer. But I think this what bagrettin interpretation. The x axis has slope of zero, it will never intersect f(x). If you increase the slope of g(x) it might. Howmuch depends on the slope of f(x) -it has to be greater the slope of f(x) otherwise the g(x) being slant will miss the aim.

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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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x can have a slope of 0 and y-intercept >=6 what will happen then? the lines should cross ... may be it's better to operate with the slope(x)>slope(x) for two functions f(x) and g(x`)
gmat1220 wrote:
I didn't get the right answer. But I think this what bagrettin interpretation. The x axis has slope of zero, it will never intersect f(x). If you increase the slope of g(x) it might. Howmuch depends on the slope of f(x) -it has to be greater the slope of f(x) otherwise the g(x) being slant will miss the aim.

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The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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Hi guys,

I am actually quite surprised by my capability to even come close to solving this 75% difficulty questions. The only thing I don't understand here is why we're looking for a solution in the 1st quadrant. For example, bagrettin refers to this in his solution.

Answer choices (C) and (D) offer solutions of $$-\frac{1}{2}$$ and $$-\frac{7}{4}$$ in the x<0 or 2nd quadrant area. Why do we not except these answers here?

Thank you!
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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1
Start by checking the slopes and intercepts of all the lines. On comparison with the reference line provided, we see that the first four lines will never intersect with the given line. Only the last line will intersect because it has a steeper slope.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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Thank you VeritasPrepKarishma! This was by far the clearest thing I've read on coordinate geometry! Thank you so much!
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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if two lines intersect we know that we can equate the two linear equation in this 2x+4=g(x).
I'm taking only positive values of x since we are looking for a intersection in the first quad.
you can easily eliminate options C,D . As for the rest a little calculation shows a,b are out . 2x+4=x-2--> x=-6 which deosnt statisfy the equation. so is B.
upon solving E 3x-2=2x+4==>x=6 which satisfies bth equations.
@@Bunuuel Can you verfiy this approach and any other concept that can be applied.
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The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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zaur2010 wrote:
Attachment:
pic.JPG
The figure above shows the graph of a function f, defined by f(x) = |2x| + 4 for all numbers x. For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ?

A. g(x) = x - 2
B. g(x) = x + 3
C. g(x) = 2x - 2
D. g(x) = 2x + 3
E. g(x) = 3x - 2

My quick analysis-

Lines A and B have slopes which are less than the slope of f(x) and will always be below the graph.

Lines C and D are parallel to F(x).

We are left with E which has more positive slope and will definitely cut the graph at some point.

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Originally posted by samusa on 01 Aug 2015, 11:26.
Last edited by samusa on 01 Aug 2015, 12:16, edited 1 time in total.
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The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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PriyankaKabbina wrote:
if two lines intersect we know that we can equate the two linear equation in this 2x+4=g(x).
I'm taking only positive values of x since we are looking for a intersection in the first quad.
you can easily eliminate options C,D . As for the rest a little calculation shows a,b are out . 2x+4=x-2--> x=-6 which deosnt statisfy the equation. so is B.
upon solving E 3x-2=2x+4==>x=6 which satisfies bth equations.
@@Bunuuel Can you verfiy this approach and any other concept that can be applied.

Couple of concepts tested in this question:

1. |x| =x for x $$\geq$$ 0 . This is used to find the equation of |2x|+4 in the 1st quadrant.

2. Slope of lines y = mx+c and y = mx + b are same.
2.1: A greater value of the slope indicates a greater angle that the line makes with the positive direction of the x-axis.

Point 1 is used to find that the equation of the line in the first quadrant will be , y =2x+4 and thus its slope will be 2.

Points 2 and 2.1 are used to eliminate options A and B. Slope of x $$\pm$$ a will be 1 and will thus be diverging away from line y =2x+4 (as slope of 1 < slope of 2, in the 1st quadrant).

Point 2 shows that lines in options C and D will be parallel to y =2x+4 ( as the slopes of [y = 2x $$\pm$$ a] = slope of [y =2x+4]). 2 parallel lines can never intersect and thus these 2 options are eliminated.

Thus E is the only remaining answer choice.

Your method is fine for this question but the method mentioned above by me can be applied to any question related to lines in xy coordinate plane.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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samichange wrote:
zaur2010 wrote:
Attachment:
pic.JPG
The figure above shows the graph of a function f, defined by f(x) = |2x| + 4 for all numbers x. For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ?

A. g(x) = x - 2
B. g(x) = x + 3
C. g(x) = 2x - 2
D. g(x) = 2x + 3
E. g(x) = 3x - 2

My quick analysis-

Lines A and B have slopes which are less than the slope of f(x) and will always be below the graph.

Lines C and D are parallel to F(x).

We are left with D which has more positive slope and will definitely cut the graph at some point.

I donot mind Kudos    I think you wanted to mention E instead of D.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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Engr2012 wrote:
samichange wrote:
zaur2010 wrote:
Attachment:
pic.JPG
The figure above shows the graph of a function f, defined by f(x) = |2x| + 4 for all numbers x. For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ?

A. g(x) = x - 2
B. g(x) = x + 3
C. g(x) = 2x - 2
D. g(x) = 2x + 3
E. g(x) = 3x - 2

My quick analysis-

Lines A and B have slopes which are less than the slope of f(x) and will always be below the graph.

Lines C and D are parallel to F(x).

We are left with D which has more positive slope and will definitely cut the graph at some point.

I donot mind Kudos    I think you wanted to mention E instead of D.

Thanks for pointing out  _________________
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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The figure above shows the graph of a function f, defined by f(x) = |2x| + 4 for all numbers x. For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ?

Understanding slope would be helpful in this question .

Standard slope eq : y=mx+b

as the slope of f(x) is 2, for g(x) to intersect, the slope of g(x) needs to be bigger than 2.
(If slope is equal to 2, then it would be parallel)

Only E has the slope bigger than 2.

A. g(x) = x - 2 --> Wrong, as slope is less than 2
B. g(x) = x + 3 --> Wrong, as slope is less than 2
C. g(x) = 2x - 2 --> Wrong, as slope is not great than 2, which makes it parallel of f(x)
D. g(x) = 2x + 3 --> Wrong, as slope is not great than 2, which makes it parallel of f(x)
E. g(x) = 3x - 2 --> CORRECT, as slope is bigger than 2.
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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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2
Clearly the option E

we can see that g(x) = 3x-2 is the line which will intersect the given y=|2x|+4 at point (6,16)
and refer to the figure .
Attachments IMG_4465.JPG [ 1.27 MiB | Viewed 13286 times ]

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Re: The figure above shows the graph of a function f, defined by f(x) = |2  [#permalink]

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