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# Functions

Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.

# 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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