Answer
Graph A represents $f(x)$ and graph $B$ represents $f'(x)$.
Work Step by Step
As the derivative $f'(x)$ represents the change in the form of the function $f(x)$, examining the graph of $f'(x)$ shows clues about the graph of $f(x)$.
Looking at these two graphs, we notice the followings:
- Both graphs are on the positive side, but one graph is falling while the other is rising. Graph A rises but more and more slowly as $x\to\infty$, while at the same time, graph $B$ falls more and more slowly as well.
Derivative shows the rate of change of a function. The fall of the graph of $f'(x)$ shows that $f(x)$ is changing a slower and slower rate.
- Graph B seems to approach infinity as $x\to0$, while graph A appears to have a vertical tangent there (though we do not see the negative side of the $x-$axis). $f(x)$ will have a vertical tangent when $f'(x)$ is not defined.
We therefore can conclude that graph A represents $f(x)$ while graph B represents $f'(x)$.
