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 \(p-q.\)
Try Part 1.
by
Nihar Mahajan
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 \(a - b.\)
by
Brian Charlesworth
\[ \begin{eqnarray} e(x)+o(x) &= & f(x) \\ e(x)+x^2& =& o(x) \\ \end{eqnarray} \] If the above two equations are true for every real \(x\) where \(e(x)\) and \(o(x)\) are any even and odd functions respectively whereas \(f(x)\) may be an ordinary function. Find \(f(2).\)
by
Sanjeet Raria
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 \(f(94)\)?
This is original.
by
siddharth bhatt
\[ \begin{eqnarray} f_1 (x)&=&x\\ f_2 (x)&=&1-x\\ f_3 (x)&=&\frac{1}{x}\\ f_4 (x)&=&\frac{1}{1-x}\\ f_5 (x)&=&\frac{x}{x-1}\\ f_6 (x)&=&\frac{x-1}{x}\\ \end{eqnarray} \]
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 \(m+n\).
by
Gautam Sharma