Let $f(x) = x^2$ and $g(x) = 2^x$ , for all real $x$ . Then the value of $f(f(g(x)) + g(f(x)))$ at $x = 1$ is

$16$

$18$

$36$

$40$

✅ Correct Option: 3

Given: $f(x) = x^2$ and $g(x) = 2^x$

We need to find $f(f(g(x)) + g(f(x)))$ at $x = 1$

CRITICAL: Notice the parentheses! We need $f(\text{something})$ where "something" $= f(g(x)) + g(f(x))$

Find $g(1)$ :

$g(1) = 2^1 = 2$

Find $f(g(1)) = f(2)$ :

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

Find $f(1)$ :

$f(1) = 1^2 = 1$

Find $g(f(1)) = g(1)$ :

Since $f(1) = 1$ , we need $g(1) = 2^1 = 2$

Calculate $f(g(1)) + g(f(1))$ :

$f(g(1)) + g(f(1)) = 4 + 2 = 6$

Apply function $f$ to this result:

$f(f(g(1)) + g(f(1))) = f(6) = 6^2 = 36$

Memory Trick: First find what's inside the outer $f(\text{here})$ , then apply $f$ to that result

Answer: $36$

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