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The figure above shows the graph of the function f defined by

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The figure above shows the graph of the function f defined by [#permalink] New post 10 Sep 2017, 11:16
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#GREpracticequestion The figure above shows the graph of the function.jpg
#GREpracticequestion The figure above shows the graph of the function.jpg [ 13.11 KiB | Viewed 1221 times ]


The figure above shows the graph of the function \(f\) defined by \(f(x) = |2x| + 4\) for all numbers x. For which of the following functions \(g\), \(f(x)\) 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. \(g(x) = 3x − 2\)

Practice Questions
Question: 3
Page: 118
Difficulty: medium
[Reveal] Spoiler: OA

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Re: The figure above shows the graph of the function f defined by [#permalink] New post 10 Sep 2017, 11:19
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You can see that all five choices are linear functions whose graphs are lines with various slopes and y-intercepts. The graph of Choice A is a line with slope 1 and y-intercept −2. It is clear that this line will not intersect the graph of f to the left of the y-axis. To the right of the y-axis, the graph of f is a line with slope 2, which is greater than slope 1. Consequently, as the value of x increases, the value of y increases faster for f than for g, and therefore the graphs do not intersect to the right of the y-axis. Choice B is similarly ruled out. Note that if the y-intercept of either of the lines in Choices A and B were greater than or equal to 4 instead of less than 4, they would intersect the graph of f. Choices C and D are lines with slope 2 and y-intercepts less than 4. Hence, they are parallel to the graph of f (to the right of the y-axis) and therefore will not intersect it. Any line with a slope greater than 2 and a y-intercept less than 4, like the line in Choice E, will intersect the graph of f (to the right of the y-axis).

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Re: The figure above shows the graph of the function f defined by [#permalink] New post 02 Nov 2018, 12:52
I took it for a little more difficult problem, with f(g(x)) to be checked if intersecting the f(x) graph
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Re: The figure above shows the graph of the function f defined by [#permalink] New post 08 Jun 2019, 08:38
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Carcass wrote:
Attachment:
#GREpracticequestion The figure above shows the graph of the function.jpg


The figure above shows the graph of the function \(f\) defined by \(f(x) = |2x| + 4\) for all numbers x. For which of the following functions \(g\), \(f(x)\) 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. \(g(x) = 3x − 2\)

Practice Questions
Question: 3
Page: 118
Difficulty: medium


--------------------------
ASIDE: Some students may be unfamiliar with the above format.
The most common way to define a line or curve is to write y = some expression involving x (e.g., y = 2x - 5)
Just know that graphing the equation y = 2x - 5, is the same as graphing the function f(x) = 2x - 5

Likewise, graphing the function f(x) = |2x|+ 4 is the SAME as graphing the equation y = |2x|+ 4
--------------------------

Let's first find the slope of one of the arms of the graph.

To do so, let's find 2 points that lie on the graph.

f(0) = |2(0)|+ 4
= |0|+ 4
= 4
So, when x = 0, y = 4
(0, 4) is one point.

f(1) = |2(1)|+ 4
= |2|+ 4
= 6
So, when x = 1, y = 6
(1, 6) is another point.

Apply the slope formula to get: slope = (6 - 4)/(1-0) = 2/1 = 2
So, the slope of the red arm is 2
Image

At this point, we might see that the graphs for answer choices C and D both have slope 2.
We know this because each is written in slope y-intercept form

For example, g(x) = 2x - 2 (aka y = 2x - 2) represents a line with slope 2 and a y-intercept of -2
Likewise, g(x) = 2x + 3 represents a line with slope 2 and a y-intercept of 3

Since both lines have the same slope of the red arm of our graph, they are PARALLEL with the red arm.
This means neither line will ever intersect the graph of f
Image
ELIMINATE C and D


Now notice that the graphs for answer choices A and B both have slope 1.
That is, g(x) = x - 2 (aka y = 1x - 2) represents a line with slope 1 and a y-intercept of -2
And g(x) = x + 3 represents a line with slope 1 and a y-intercept of 3
Since both lines have a slope that's LESS THAN 2, both lines will diverge away from the red arm.
So, neither line will ever intersect the graph of f
Image
ELIMINATE A and B

By the process of elimination, the correct answer is E, but let's check it out for "fun"
The graph for answer choice E (y = 3x - 2) has slope 3.
Since the slope of y = 3x - 2 is steeper than the red arm of the graph, we know that the lines will intersect at some point.
Image

Answer: E

Cheers,
Brent
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Re: The figure above shows the graph of the function f defined by   [#permalink] 08 Jun 2019, 08:38
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The figure above shows the graph of the function f defined by

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