Suppose we are given the functions:
$f(x) = x$
$g(x) = x^2$
$h(x) = \sin(x)$
$m(x) = 2^x$
$n(x) = log_2(x)$
We are then asked to express each graph as the addition, subtraction, multiplication or division of a pair of the functions above.
Here are the graphs of such functions.
Is there any way to easily tell which two functions are combined together (eg. wavy graphs must contain a $h(x) = \sin(x)$)?
If you look at the first graph you can see that $f(0) = -1$. Now we can see that when $x = 0$, that the functions $x$, $x^2$, and $sin(x)$ are equal to zero, and $log_2(x)$ is undefined. This tells us that we must subtract $2^x$ from something (we must subtract since $2^x$ is $1$). Notice that as $x \rightarrow -\infty$ that $2^x \rightarrow 0$. Coming from $\infty$ and moving to the right that the graph is essentially linear. This tells us that $f(x) = x - 2^x$.
Notice that the second graph is periodic. This tells us that there has to be $sin(x)$ in this equation. Also notice that the amplitude is steadily increasing as $x \rightarrow \infty$. This tells us that it is being multiplied by something. Since the maximum value that $sin(x)$ can take on is $1$, the value of the peaks corresponds to the value of the function that is multiplied by x evaluated at that point. By seeing that $g(x) = 1000$ when $x$ is a little greater than $30$, we can eliminate the functions $x$ (because it grows too slowly), and $2^x$ (because it grows too quickly). The only function whose growth is between these is $x^2$. So the second graph is $x^2sin(x)$.
In the last graph, there are several things that stick out. The first to notice is that the graph is not defined for $x < 0$. Note that $log_2(x)$ is not defined for $x \leq 0$. So there must be a $log_2(x)$ term. Also note that $log_2(x) \rightarrow -\infty$ as $x \rightarrow 0$. If we multiply this by a function that $\rightarrow 0$ as $x \rightarrow 0$ then we can achieve the shape in the graph. So our choices are $x, x^2,$ and $sin(x)$. Note that $h(x) \approx 4$ when $x = 2$. Given that $log_2(2) = 1$ and $x^2 = 4$ we try $x^2log_2(x)$. Checking a few more values confirms this.