Algebra

Functions

Functions: Level 2 Challenges

         

Consider a function ff satisfying

f(x)=x2.f\big(\sqrt{x}\big)=x^2.

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

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

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

xy=1x+1y,x#y=x+yxyx * 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 kk such that

(22k)#(k33)=27.(22 * k) \# (k * 33) = 27.

If ff is a function such that f(f(x))=x21 f(f(x)) = x^2 - 1 , what is the value of f(f(f(f(3)))) f(f(f(f(3)))) ?

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