If \(f(x)=x^x,\) then \(f(f(x))=\)

[Gmat_mission]

If \(f(x)=x^x,\) then \(f(f(x))=\)

A. \(x^{x^{x^x}}\)
B. \(x^{x^{x}}\)
C. \(x^{x^{x^2}}\)
D. \(x^{x^{x+1}}\)
E. \((x^x)^x\)

[spoiler]OA=D[/spoiler]

Source: Veritas Prep

[Scott@TargetTestPrep]

If \(f(x)=x^x,\) then \(f(f(x))=\)

A. \(x^{x^{x^x}}\)
B. \(x^{x^{x}}\)
C. \(x^{x^{x^2}}\)
D. \(x^{x^{x+1}}\)
E. \((x^x)^x\)

[spoiler]OA=D[/spoiler]

Source: Veritas Prep

Solution:

Since f(x) = x^x, f(f(x)) = f(x^x) = (x^x)^(x^x) = x^(x * x^x) = x^(x^(x + 1)).

Alternate Solution:

Let’s let x = 2. Then, f(2) = 2^2 = 4 and f(f(2)) = f(4) = 4^4 = 2^8. Let’s test each answer choice to see which one(s) simplify to 2^8:

A) x^x^x^x

If x = 2, we get 2^2^2^2 = 2^2^4 = 2^16. A is not correct.

B) x^x^x

If x = 2, we get 2^2^2 = 2^4. B is not correct.

C) x^x^x^2

If x = 2, then we get 2^2^2^2 which is the same as answer choice A. C is not correct.

D) x^x^(x + 1)

If x = 2, then we get 2^2^(2 + 1) = 2^2^3 = 2^8. D might be correct.

E) (x^x)^x

If x = 2, we get (2^2)^2 = 4^2 = 2^4. E is not correct.

Since D is the only answer choice that yields 2^8 when x is 2, D must be the correct answer choice.

Answer: D