To find the value of \( f(g(x)) \), you need to first determine \( g(x) \) and then plug it into \( f(x) \).
1. Calculate \( g(x) \):
Given \( g(x) = \sqrt{\sqrt{x}} \)
For \( x \geq 0 \), we start with \( g(x) = \sqrt{x} \) and then take the square root of that result.
So, \( g(x) = \sqrt{x} = x^{1/2} \)
Now, taking the square root again gives: \( g(x) = \sqrt{x^{1/2}} = x^{1/4} \)
2. Substitute \( g(x) \) into \( f(x) \):
Now that we know \( g(x) = x^{1/4} \), we can find \( f(g(x)) = f(x^{1/4}) \).
Substitute \( x^{1/4} \) into \( f(x) = 16x^2 \):
\( f(g(x)) = 16(x^{1/4})^2 = 16x^{2/4} = 16x^{1/2} = 16\sqrt{x} \)
Therefore, for \( x \geq 0 \), the value of \( f(g(x)) \) is \( 16\sqrt{x} \).
To find the value of \( g(f(x)) \), we follow a similar process.
1. Calculate \( f(x) \):
Given \( f(x) = 16x^2 \)
Now we plug \( f(x) \) into \( g(x) = \sqrt{\sqrt{x}} \):
\( g(f(x)) = \sqrt{\sqrt{16x^2}} = \sqrt{\sqrt{(4x)^2}} = \sqrt{\sqrt{4^2x^2}} = \sqrt{\sqrt{16x^2}} = \sqrt{4x} = 2\sqrt{x} \)
Therefore, for \( x \geq 0 \), the value of \( g(f(x)) \) is \( 2\sqrt{x} \).
Finally, for functions to be inverse functions, they must "undo" each other. In this case, \( f(x) \) and \( g(x) \) are not inverse functions because composing them in either order doesn't result in the identity function \( f(g(x)) = x \) or \( g(f(x)) = x \).
