Functions: Level 4 Challenges


Andrew has a favorite function A(x)=px+qxA(x)=px+q^x such that A(1)=4A(1)=4 and 4A(2)=374A(2)=37, find the maximum value of pq.p-q.

Try Part 1.

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

If f(8)=3f(8) = 3 then log9(f(2015))=ab\log_{9}(f(2015)) = \dfrac{a}{b}, where aa and bb are positive coprime integers. Find ab.a - b.

e(x)+o(x)=f(x)e(x)+x2=o(x) \begin{aligned} e(x)+o(x) &= & f(x) \\ e(x)+x^2& =& o(x) \\ \end{aligned} If the above two equations are true for every real xx where e(x)e(x) and o(x)o(x) are any even and odd functions respectively whereas f(x)f(x) may be an ordinary function. Find f(2).f(2).

The lovely function ff_{}^{} has the beautiful property that, for each real number xx:

f(x)+f(x1)=x2.\large f(x)+f(x-1) = x^2.

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


This is original.

f1(x)=xf2(x)=1xf3(x)=1xf4(x)=11xf5(x)=xx1f6(x)=x1x \begin{aligned} 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{aligned}

If we know that f6(fm(x))=f4(x) f_6 (f_m(x))=f_4(x) and fn(f4(x))=f3(x) f_n (f_4(x))=f_3(x) , find the minimum value of m+nm+n.


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