Algebra
# Functions

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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