If g(x) = 2^x + x, how many solutions satisfy the equation g(x) = 2?

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.

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 \(2^x+x=2 \implies 2^x=2-x\).

We will draw the graph of \(y=2^x\) and \(y=2-x\)

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.


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.

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.

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!

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