Question 2
Consider the following functions on the nonzero real numbers:
$$P(x)=(x^2-1)^{2023},\qquad Q(x)=(2x+1)^{14},\qquad R(x)=\left(2x+1+\frac2x\right)^{34}.$$
Initially a list contains exactly two of these three functions. At each operation, one may add, subtract or multiply functions in the list, or raise a listed function to a positive integer power. One may also add, subtract or multiply a listed function by any real constant, obtaining a new function and adding it to the list. These operations may be repeated. Prove that, whichever two functions are chosen initially, the remaining function cannot be obtained.
$$P(x)=(x^2-1)^{2023},\qquad Q(x)=(2x+1)^{14},\qquad R(x)=\left(2x+1+\frac2x\right)^{34}.$$
Initially a list contains exactly two of these three functions. At each operation, one may add, subtract or multiply functions in the list, or raise a listed function to a positive integer power. One may also add, subtract or multiply a listed function by any real constant, obtaining a new function and adding it to the list. These operations may be repeated. Prove that, whichever two functions are chosen initially, the remaining function cannot be obtained.