4. We say that the constant $a$ is a fixed point of a function $f$ if $f(a)=a$. Find all values of $c$ such that $f(x)=x^{2}-2$ and $g(x)=2 x^{2}-c$ share a common fixed point.
Answer: 3 and 6
Solution: Note that if $f(x)=x$, then $x^{2}-x-2=0$, or $(x-2)(x+1)=0$. Thus, the fixed points of $f$ are 2 and -1 , and so we wish for either of these to be a fixed point of $g$. If 2 is a fixed point of $g$, we must have $g(2)=2=8-c$, and so $c=6$. If -1 is a fixed point of $g$, then $2-c=-1$, and $c=3$.