If $$ u(t) = 4t - 5 $$, $$ v(t) = t^2 $$, and $$ f(t) = \frac{1}{t} $$, find the formula for $$ u(f(v(t))) $$.

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To determine the formula for $$ u(f(v(t))) $$, we evaluate the composite functions step by step:

**Step 1: Evaluate $$ v(t) = t^2 $$**  
Substitute $$ v(t) = t^2 $$.

**Step 2: Evaluate $$ f(v(t)) = f(t^2) $$**  
Since $$ f(t) = \frac{1}{t} $$, we substitute $$ t^2 $$:  
$$
f(t^2) = \frac{1}{t^2}
$$

**Step 3: Evaluate $$ u(f(v(t))) = u\left(\frac{1}{t^2}\right) $$**  
Substitute into $$ u(t) = 4t - 5 $$:  
$$
u\left(\frac{1}{t^2}\right) = 4\left(\frac{1}{t^2}\right) - 5
$$  
Simplify:  
$$
u(f(v(t))) = \frac{4}{t^2} - 5
$$

Thus, the formula for $$ u(f(v(t))) $$ is **$$ \frac{4}{t^2} - 5 $$**.

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