Functions: Level 3 Challenges


Define a function f:RRf : \mathbb{R} \to \mathbb{R} such that f(f(x))=x2x+1f(f(x))=x^2-x+1 for all real xx.

Evaluate f(0)=f(0) = \, \text{? } .

If f(x2015+1)=x4030+x2015+1f(x^{2015}+1)=x^{4030}+x^{2015}+1, then what is sum of the coefficients of f(x20151)?f(x^{2015}-1)?

f(x) f(1x)=f(x)+f(1x)\large f(x) \ f\left(\dfrac 1 x\right) = f(x) + f\left(\dfrac 1 x\right)

A polynomial ff satisfies the above equation and f(10)=1001.f(10) = 1001. Find the value of f(20).f(20).

f(x)=9x9x+3\large f(x)=\frac{9^x}{9^x+3}

Suppose we define f(x)f(x) as above. Let a=f(x)+f(1x)a=f(x)+f(1-x) and b=f(11996)+f(21996)+f(31996)++f(19951996).b=f\left(\frac1{1996}\right) + f\left(\frac2{1996}\right) + f\left(\frac3{1996}\right)+\ldots+ f\left(\frac{1995}{1996}\right).

Evaluate a+ba + b.

If f(x)=ax+bf\left( x \right) =ax+b, where aa and bb are real numbers, and f(f(f(x)))=8x+21f\left( f\left( f\left( x \right) \right) \right) =8x+21, what is a+ba+b?


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