Functions: Level 3 Challenges


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

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

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

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

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

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

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

Evaluate \(a + b\).

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


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