Author 
Message 
TAGS:

Hide Tags

Senior Manager
Joined: 20 Aug 2015
Posts: 391
Location: India

If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
07 Feb 2016, 19:10
Question Stats:
31% (01:56) correct 69% (01:38) wrong based on 290 sessions
HideShow timer Statistics
If \(g(x) = 2^x + x\), how many solutions satisfy the equation g(x) = 2? A. Zero B. One C. Two D. Three E. Infinite
Official Answer and Stats are available only to registered users. Register/ Login.



Current Student
Joined: 20 Mar 2014
Posts: 2633
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
07 Feb 2016, 19:20
TeamGMATIFY wrote: If g(x) = \(2^x\) + x, how many solutions satisfy the equation g(x) = 2?
A. Zero B. One C. Two D. Three E. Infinite The number of solutions of 2^x+x = 2 will be equal to number of points of intersection of y=2 and y=2^x+x. Realize that 2^x+x is exponential in nature and as such once it passes the x=0.641, the value of y=2^x+x will go on increasing on an exponential basis (note that it is NOT necessary to know this value of 'x' but you do need to know the general nature of an exponential function). Thus, there will only be 1 point of intersection. B is thus the correct answer. Refer below for the graph. Attachment:
20160207_211951.jpg [ 31.64 KiB  Viewed 3169 times ]



Intern
Joined: 14 Jul 2015
Posts: 22
GMAT 1: 680 Q44 V40 GMAT 2: 710 Q49 V37

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
Updated on: 07 Feb 2016, 20:49
Usually when I see problems like this (graph, number of times a value is hit, exponential equation, etc.), I try to find a key inflection x value. For any a^b question, the inflection x will often be 0. Be careful with this rule of thumb, of course. Don't trust it blindly. Once we've defined our inflection point (x = 0 for this question), we break our equation into three parts: x = 0, x < 0, x > 0. We want to know what the function is doing in each of these ranges. For this specific question, we want to know if each part is upwardtrending or downwardtrending.
x = 0: 2^0 + 0 = 1. g(1) = 1. Next question: are both x < 0 and x > 0 upward trending? Downward trending? Etc.
x < 0: \(2^x + x, x < 0\) For simplicity, let's flip the signs on everything so we're not dealing with awkward negatives to throw us off: \(\frac{1}{{2^x}}  x, x > 0\). This means that, returning to x < 0, the lesser x is, the more negative y is. The only unsurety is for 1 < x < 0. If you know calculus and limits, \(\frac{1}{{2^x}} + x\) as x approaches 0 will become 1/1 + 0, which is 1. We conclude that the trend is strictly downwards.
x > 0: \(2^x + x, x > 0\) \(2^x\) is a hyperbola when x > 0. We know that it will pass through all positive x points and, for x > 0, all y points above 1. Our trend is strictly upwards.
Since g(1) = 1, g(x<0) < 1, g(x>0) > 1, x>0 is the upward trending half of a hyperbola, and x>0 will cover the case of g(1) = 1, we know that there will be only one point where g(x) crosses 2.
Edit: Added a warning about x = 0 as an inflection point.
Originally posted by Beixi88 on 07 Feb 2016, 20:33.
Last edited by Beixi88 on 07 Feb 2016, 20:49, edited 2 times in total.



Current Student
Joined: 20 Mar 2014
Posts: 2633
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
07 Feb 2016, 20:42
Beixi88 wrote: Usually when I see problems like this (graph, number of times a value is hit, exponential equation, etc.), I try to find a key inflection x value. For any a^b question, the inflection x will almost always be 0. From here, we break our equation into three parts: x = 0, x < 0, x > 0. We want to know what the function is doing in each of these ranges. For this specific question, we want to know if each part is upwardtrending or downwardtrending.
x > 0: \(2^x + x, x > 0\) \(2^x\) is a parabola when x > 0. We know that it will pass through all positive x points and, for x > 0, all y points above 1. Our trend is strictly upwards.
IMO, statement above in red is a sweeping statement that is not really true. "Almost always" is a bit misleading. Secondly, the graph of y=2^x is NOT a parabola but one half of a hyperbola with x axis as the asymptotic axis after \(x \approx 11\)



Intern
Joined: 14 Jul 2015
Posts: 22
GMAT 1: 680 Q44 V40 GMAT 2: 710 Q49 V37

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
07 Feb 2016, 20:46
Engr2012 wrote: Beixi88 wrote: Usually when I see problems like this (graph, number of times a value is hit, exponential equation, etc.), I try to find a key inflection x value. For any a^b question, the inflection x will almost always be 0. From here, we break our equation into three parts: x = 0, x < 0, x > 0. We want to know what the function is doing in each of these ranges. For this specific question, we want to know if each part is upwardtrending or downwardtrending.
x > 0: \(2^x + x, x > 0\) \(2^x\) is a parabola when x > 0. We know that it will pass through all positive x points and, for x > 0, all y points above 1. Our trend is strictly upwards.
IMO, statement above in red is a sweeping statement that is not really true. "Almost always" is a bit misleading. Secondly, the graph of y=2^x is NOT a parabola but one half of a hyperbola with x axis as the asymptotic axis after \(x \approx 11\) Thanks for pointing out those errors. I'll go ahead and amend my post.



Senior CR Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1381
Location: Viet Nam

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
17 May 2017, 09:38
TeamGMATIFY wrote: If g(x) = \(2^x\) + x, how many solutions satisfy the equation g(x) = 2?
A. Zero B. One C. Two D. Three E. Infinite This question is too hard and unlikely to appear on the GMAT test. We could draw the graph as Engr2012 did. However, drawing the graph of \(2^x+x\) is too hard without any tool. Note that \(2^x+x=2 \implies 2^x=2x\). We will draw the graph of \(y=2^x\) and \(y=2x\) Graph of \(y=2^x\) If \(x=0 \implies y=2^x = 1\). The line go through point (0, 1) If \(x=1 \implies y=2^x = 2\). The line go through point (1, 2) If \(x=2 \implies y=2^x = 4\). The line go through point (2, 4) If \(x=1 \implies y=2^x = 1/2\). The line go through point (1, 1/2) Hence, the graph of \(y=2^x\) looks like the blue line. Attachment:
graph.png [ 10.67 KiB  Viewed 2069 times ]
Hence, the equation has only 1 root. The answer is B. We could solve this question using derivative. However, this tool is out of scope.
_________________
Actual LSAT CR bank by Broall
How to solve quadratic equations  Factor quadratic equations Factor table with sign: The useful tool to solve polynomial inequalities Applying AMGM inequality into finding extreme/absolute value
New Error Log with Timer



Current Student
Joined: 22 Sep 2016
Posts: 181
Location: India
GPA: 4

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
11 Jul 2017, 08:57
TeamGMATIFY wrote: If g(x) = \(2^x\) + x, how many solutions satisfy the equation g(x) = 2?
A. Zero B. One C. Two D. Three E. Infinite Bunuel I am not able to comprehend the explanations. Is there an algebraic approach to this question? I don't know about graphs, at all.
_________________
Desperately need 'KUDOS' !!



GMAT Tutor
Joined: 24 Jun 2008
Posts: 1327

If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
11 Jul 2017, 09:58
I agree with the post above that says this question is out of the scope of the GMAT, since you can't solve the resulting equation using conventional GMAT math, so I don't think people should worry about it much. But I suppose you can answer it without using any graphs. If you think about the function f(x) = 2^x, that is a function that is 'constantly increasing'. If you plug in bigger and bigger values for x, the value of the function gets bigger and bigger. The same is true for the function h(x) = x, and if we add two functions that are constantly increasing, the result will be too. So the function g(x) = 2^x + x is a function that gets bigger and bigger as you plug in larger and larger values of x. Put algebraically, if a > b, then g(a) > g(b) and as a logical consequence, if we plug in two different values for x into g(x), we can never get the same answer. So the equation g(x) = 2 can only have at most one solution for x. Then we just need to confirm there is indeed one solution. This is where things really get outside the scope of the GMAT. If you notice that g(x) can be less than 2 (if you plug in x = 0, say) or greater than 2 (if you plug in x=1, say), then g(x) has to be equal to 2 for some value of x between 0 and 1, because g(x) is constantly and continuously increasing, and has to pass through the value 2 at some point. Technically I'm using something called the 'Intermediate Value Theorem' here though, which is a theorem from calculus that you don't ever need on the GMAT. The logic above might not make a lot of sense to people who haven't studied calculus (you use this kind of reasoning in a lot of calc problems, but rarely in other kinds of problems), and as I pointed out above, if it doesn't make a lot of sense, it's not something you need to worry about if you're preparing for the GMAT!
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com



NonHuman User
Joined: 09 Sep 2013
Posts: 8514

Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?
[#permalink]
Show Tags
23 Sep 2018, 14:24
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2? &nbs
[#permalink]
23 Sep 2018, 14:24






