i would like to clarify some questions from GRE,which at first seems a little difficult to understand,suppose that we have some function
$f(x)=|2*x|+4$
and graph of this function is given
where is following question: For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ? and 4 possible answers are
A.g(x)=x-2
B.g(x)=x+3
C.g(x)=2*x-2
D.g(x)=2*x+3
E.g(x)=3*x-2
first what i did not understand what does mean
For which of the following functions g defined for all numbers x does the graph of g intersect the graph of f ? does it means that which g intersect of f for all x or?answer is E,i have guessed that for intersection we should have $|2*x|+4=g(x)$,clearly C and D no,because they are parallel lines with common slopes,so whe should have
$|2*x|+4=x-2$
or $|2*x|+4=x+3$
or $|2*x|+4=3*x-2$
after some calculation we will get $|2*x|=x-6$
or $|2*x|=x-3$
|2*x|=3*x-6
first can't be ,because $|2*x|>=x$ second also using the same rule,only one left
$|2*x|=3*x-6$ by using solving,for example if $x>0$ $2*x=3*x-6$ $x=6$
if $x<0$
$-2*x=3*x-6$
$x=6/5$
is there any short way to solve it?
You only need the two graphs to intersect at one point. If this is so for a single value of $x$, then the corresponding item will be a solution.
You are correct that "E" is the only answer. A quick way to solve the problem is to note that the graphs of all the items are straight lines; so you can deduce things from geometric reasoning. Just look at the $y$-intercepts of the lines and compare the slopes of the lines to the slopes of the line segments comprising the graph of $f$. It would also be helpful, for each item, to separate the problem into two, natural, parts: Does the given line intersect the graph of $f$ at some $x\ge0$? Does the given line intersect the graph of $f$ at some $x<0$?
For example:
E: The $y$ coordinate of the $y$-intercept of the line here is $-2$, which is less than $f(0)=4$, and the slope is $3$. Since the slope of the line segment given by $y=f(x), x\ge 0$ is $2<3$, the graph of $y=g(x)$ will intersect the graph of $y=f(x)$ for some $x>0$ (over $[0,\infty)$, the graph of $g$ rises more quickly than the graph of $f$).
If you sketch the graphs of the given lines (which you should become adept at for the GRE), paying close attention to the slope of the line, you can easily see what's happening.