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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 2 Challenges

Consider a function $$f$$ satisfying

$f\big(\sqrt{x}\big)=x^2.$

What is the value of $$f(2) ?$$

Suppose $$f$$ is a real function satisfying $$f(x+f(x)) = 4f(x)$$ and $$f(1)=4$$. What is $$f(21)$$?

Given that $$f(2^x)+xf(2^{-x})=1$$, find the value of $$f(2)$$.

$x * y = \frac{1}{x} + \frac{1}{y}, \quad x \# y = \frac{x+y}{x-y}$

Let the operations $$\#$$ and $$*$$ be defined as described above.

Find the value of $$k$$ such that

$(22 * k) \# (k * 33) = 27.$

If $$f$$ is a function such that $$f(f(x)) = x^2 - 1$$, what is the value of $$f(f(f(f(3))))$$?

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