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Algebra

Functions

Functions: Level 4 Challenges

         

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.

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 \(a - b.\)

\[ \begin{eqnarray} e(x)+o(x) &= & f(x) \\ e(x)+x^2& =& o(x) \\ \end{eqnarray} \] 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 \(f(2).\)

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 \(f(94)\)?


Inspiration.

This is original.

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

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 \(m+n\).

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