<Question>
If $t(x) = 3-g(x)$ and $g(x) = \sqrt{x}$, then what is $t(g(16))$?
</Question>

<think>
<Plan_1>
Okay, let me try to figure out how to solve this problem. The question is asking for t(g(16)), given that t(x) is defined as 3 minus g(x), and g(x) is the square root of x. Hmm, okay. Let me break this down step by step.

First, I need to understand what each function does. The function g(x) takes an input x and returns the square root of that input. So, for example, if x is 16, then g(16) would be the square root of 16, which is 4.

Then, the function t(x) is defined as 3 minus g(x). Wait, hold on. Let me make sure I parse this correctly. The problem says t(x) = 3 - g(x). So t is a function that takes an input x, subtracts the value of g(x) from 3. But here, we need to find t(g(16)). That means we first apply g to 16, and then apply t to the result of that.

So, let's start with the inner function first. Let's compute g(16).
Since g(x) is sqrt(x), g(16) = sqrt(16) = 4. Got that. So g(16) is 4.

Now, we need to compute t(g(16)), which is t(4). Because g(16) is 4, so t of that is t(4).

<prompt_1.1> Calculate t(g(16)) using the above plan. </prompt_1.1>
-----

Wait, but let me verify this again to make sure I didn't mix up the functions. Sometimes, when functions are composed, the order can be tricky. Let me check the definitions again.

<prompt_1.2> Re-evaluate the functions and verify the solution using the above plan. </prompt_1.2>
-----

But just to be thorough, let me consider if there's another way to approach this problem. Maybe by combining the functions first? Let's see.

<prompt_1.3> Explore an alternative method by combining functions using the above plan. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
Now, t(x) is 3 - g(x). Wait a second, here t is also a function that takes x as input, but inside t(x), it subtracts g(x) from 3. But here, since we're evaluating t at 4, we need to compute 3 minus g(4). Because in t(x), the expression is 3 - g(x), so substituting x with 4 gives us 3 - g(4).

So, t(4) = 3 - g(4). Now, g(4) is sqrt(4), which is 2. Therefore, t(4) = 3 - 2 = 1.
</execution_1.1>

<execution_1.2>
The function t(x) is given as 3 - g(x). So, when we evaluate t at some value, say a, it's 3 minus g(a). Therefore, t(a) = 3 - g(a). So, if we need t(4), that would be 3 - g(4). Then, g(4) is sqrt(4) = 2. Therefore, t(4) = 3 - 2 = 1.
</execution_1.2>

<execution_1.3>
If I try to write t(g(x)) as a composite function, that might help. So, t(g(x)) would be t(sqrt(x)). But t(x) is 3 - g(x), so substituting sqrt(x) into t, we get t(sqrt(x)) = 3 - g(sqrt(x)). But g(sqrt(x)) is sqrt(sqrt(x)), which is the fourth root of x. So t(g(x)) = 3 - x^{1/4}. Therefore, t(g(16)) would be 3 - 16^{1/4}. Let's compute 16^{1/4}.

The fourth root of 16 is 2, since 2^4 = 16. Therefore, 3 - 2 = 1. So this method also gives 1. That matches the previous result.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but let me double-check to make sure I didn't skip any steps or make a mistake in substitution. Let's go through it again.

<prompt_2.1> Re-evaluate the steps to verify the solution using the above plan. </prompt_2.1>
-----

Based on execution_1.3:
Therefore, regardless of the approach—whether substituting step by step or combining the functions first—the result is 1. 

I think that's thorough.
I considered both possible approaches, checked each step, and verified the results. No steps seem to be missing or miscalculated.
</Plan_2>

<Execution_2>
<execution_2.1>
First, compute g(16): sqrt(16) = 4. Then, compute t(4): 3 - g(4). Since g(4) is sqrt(4) = 2, so 3 - 2 = 1.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, $t(g(16)) = \boxed{1}$.
</Final_answer>
</think>