Cute Function

Method 1 : Find the value of $x$ satisfying $2^x = 16$ .

Short version:
We can write our working as $f(16) = f(2^4) = 4^2 = \boxed{16}$ .

Long version:
We are given the function $f$ which maps $2^x$ to $x^2$ , and we want to find the image that 16 maps to.

Since we are given the numerical value of 16 instead of the number $2^x$ , we must first find the unknown value of $x$ such that $2^x = 16$ .

Notice that $2^x = 16 = 2^4$ , thus by comparing the powers, we have $x = 4$ .

This leaves us with finding the image which maps $2^x$ to $x^2$ at $x=4$ . In other words, we want to find the value of $x^2$ when $x=4$ , and this gives us $\boxed{16}$ as the answer.

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Method 2 : Determine the function of $f(x)$ first.

We are given that $f(2^x) = x^2$ . Let $g(x) = 2^x$ . And so our the equation in the problem becomes $f(g(x)) = x^2$ .

If we replace $x$ with the inverse function, $g^{-1}(x)$ , then the left hand side will be simplified to just $f(g(g^{-1}(x))) = f(x)$ , and the right hand side will be simplified to $(g^{-1}(x))^2$ . In other words, we have

$f(g(x)) = x^2 \Rightarrow f(x) = (g^{-1}(x))^2 .$

And because we want to find $f(16)$ , we substitute $x=16$ to get $(g^{-1}(16))^2$ , thus what's left is to evaluate $g^{-1}(16)$ , or equivalently $g^{-1}(x)$ at $x=16$ .

Since $g(x) = 2^x$ , then the inverse function of $g(x)$ is just $g^{-1}(x) = \log_{2} (x)$ .

Hence our answer is $f(16) = ( g^{-1}(16))^2 = ( \log_{2} 16)^2 = 4^2 = \boxed{16}$ .

$f$ ( $16$ ) = $f$ ( $2^4$ ) = $4^2$ = $16$

One such function is f(x) = [log 2 (x)]^2. Hence, f(16) = [log 2 (16)]^2 = [log_2(2^4)]^2 = 4^2 = 16.

$f\left( { 2 }^{ x } \right) ={ x }^{ 2 }$

$x=4$

$f\left( { 2 }^{ 4 } \right) ={ 4 }^{ 2 }=16$ ,