graphing-functions
I have tried looking for tutorials or guides online, but I keep finding problems that are fairly similar but not exactly what I am looking for. I know how to add functions regularly, I am familiar with graphing an equation using its slope and $y$-intercept, etc., but I have no idea where to begin on finding the sum of two functions using this graph:
$$\mathrm{Use\ the \ graph \ to \ find \ }f(0)+g(0).$$
Taking logarithms can be useful because of the fact that if $x$ and $y$ are positive real numbers, then $x<y$ if and only if $\log x<\log y$, and sometimes the logarithms are easier to compare. For example, basic properties of the logarithm tell us that $\log n^2 = 2\log n$ and $\log 2^n=n\log 2$. Now look at a graph of $y=2\log x$: it rises as $x$ increases, but it also gets flatter and flatter, meaning that it’s rising more and more slowly. What about the graph of $y=(\log 2)x$? Does it get flatter as it rises, or does it keep rising at a constant slope? Which one will eventually get on top and stay there? In fact, it won’t just stay on top: it’ll keep getting further and further above the other one, so it must be growing faster.
That should help you with the comparison of $n^2$ and $2^n$. The comparison of $n^2$ with $n^2\log n$ is easier. Just look at the ratio of the two, $$\frac{n^2\log n}{n^2}=\log n\;.$$ As $n$ increases, what happens to that ratio? Does it get smaller, approach some limiting value, or get bigger? If it gets bigger, you know that the numerator, $n^2\log n$, must be growing faster than the denominator, $n^2$. If it’s roughly constant, they must be growing at about the same rate. And so on.
That leaves the comparison between $n^2\log n$ and $2^n$. Logarithms can again help here: $\log(n^2\log n)=\log n^2+\log(\log n)=2\log n+\log(\log n)$, again using basic properties of the logarithm. How does that compare with $n\log 2$, the logarithm of $2^n$, when $n$ is very large?
It appears that your graph was generated using Desmos.
When you click on the solid straight line at the point $x=2$, Desmos does explicitly say that the point is undefined, and exhibits it as a hollow “excluded point” (removable singularity).
I'm unfamiliar with why Desmos shows the exclusion only upon clicking. Maybe there is a software setting that can display them, by default.
Best Answer
Add the $y-$ values of each of the points, while keeping the $x-$values the same to get the new point for the function $(f+g)(x)$.
One thing I did not see mentioned is that the domains do not coincide. The function $(f+g)(x) = f(x)+g(x)$ can not exist where one of them does not. The domain of the function $(f+g)(x)$ is the intersection of the two domains (or "overlap"), and domain of $f$ and $g$ intersect on $D_f\cap D_g = [-4,3]$