$x!=x$

This question is basically asking, "how many times does the graph $x=y$ intersect the graph $x!=y$ ?". For positive values of $x$ , the graph looks somewhat like an exponential function. For negative integer values of $x$ , $x!$ is undefined because of a division by zero error. When this happens, this creates an asymptote in the graph. Because there are infinite negative integers, there are infinite asymptotes in the graph of $x!=y$ . However, to figure out exactly what the graph looks like, we need to test some values. Using the gamma function (or a calculator), we can calculate the values of $x!$ at fractional negative values. For example, $(-0.5)!$ is approximately equal to 1.8. We can make a table containing some inputs: $\begin{bmatrix} x & y \\ -0.5 & 1.8 \\ -1.5 & -3.5 \\ -2.5 & 2.4 \\ -3.5 & -0.9 \\ -4.5 & 0.3 \end{bmatrix}$ Based on these values, we can tell that fractional negative values of $x!$ alternate between being positive and negative, getting closer and closer to 0. However, if they're alternating between asymptotes, this means that the direction the graph of $x!$ comes from alternates as well. This means for every other asymptote, the part of the graph directly to the left of it is negative, extending downward infinitely. Since $y$ in $x=y$ decreases as $x$ decreases, there are infinite values where $x!=x$ .

Of course, this isn't a full proof, but you should be able to calculate enough values to determine that $x!=x$ is true more times than the largest answer available. Therefore, the answer is "None of these".