Algebra
# Functions

Andrew has a favorite function \(A(x)=px+q^x\) such that \(A(1)=4\) and \(4A(2)=37\), find the maximum value of \(p-q.\)

Suppose \(f\) is a continuous, positive real-valued function such that \(f(x + y) = f(x)f(y)\) for all real \(x,y.\)

If \(f(8) = 3\) then \(\log_{9}(f(2015)) = \dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a - b.\)

The lovely function \(f_{}^{}\) has the *beautiful* property that, for each real number \(x\):

\[\large f(x)+f(x-1) = x^2.\]

If \(f(19)=94\), what is \(f(94)\)?

\[ \begin{eqnarray} f_1 (x)&=&x\\ f_2 (x)&=&1-x\\ f_3 (x)&=&\frac{1}{x}\\ f_4 (x)&=&\frac{1}{1-x}\\ f_5 (x)&=&\frac{x}{x-1}\\ f_6 (x)&=&\frac{x-1}{x}\\ \end{eqnarray} \]

If we know that \( f_6 (f_m(x))=f_4(x) \) and \( f_n (f_4(x))=f_3(x) \), find the minimum value of \(m+n\).