If `f(t)=t^(2)+2 and g(t)= (1//t) +2`, then for
`t=2, f[g(t)] -g[f(t)]=`?

Options

A. 1.2
B. 2.6
C. 4.34
D. None of these

Answer

To solve the problem step by step, we need to compute \( f[g(t)] - g[f(t)] \) for \( t = 2 \).

### Step 1: Calculate \( g(t) \)
Given \( g(t) = \frac{1}{t} + 2 \), we substitute \( t = 2 \):
\[ g(2) = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]

### Step 2: Calculate \( f[g(t)] \)
Now we need to find \( f(g(2)) = f\left(\frac{5}{2}\right) \).
The function \( f(t) = t^2 + 2 \), so:
\[ f\left(\frac{5}{2}\right) = \left(\frac{5}{2}\right)^2 + 2 = \frac{25}{4} + 2 = \frac{25}{4} + \frac{8}{4} = \frac{33}{4} \]

### Step 3: Calculate \( f(t) \)
Next, we calculate \( f(2) \):
\[ f(2) = 2^2 + 2 = 4 + 2 = 6 \]

### Step 4: Calculate \( g[f(t)] \)
Now we need to find \( g(f(2)) = g(6) \):
\[ g(6) = \frac{1}{6} + 2 = \frac{1}{6} + \frac{12}{6} = \frac{13}{6} \]

### Step 5: Calculate \( f[g(t)] - g[f(t)] \)
Now we can find the difference:
\[ f[g(2)] - g[f(2)] = \frac{33}{4} - \frac{13}{6} \]

### Step 6: Find a common denominator and compute the difference
The least common multiple of 4 and 6 is 12. We convert both fractions:
\[ \frac{33}{4} = \frac{33 \times 3}{4 \times 3} = \frac{99}{12} \]
\[ \frac{13}{6} = \frac{13 \times 2}{6 \times 2} = \frac{26}{12} \]
Now we subtract:
\[ \frac{99}{12} - \frac{26}{12} = \frac{99 - 26}{12} = \frac{73}{12} \]

### Final Answer
Thus, the final answer is:
\[ f[g(2)] - g[f(2)] = \frac{73}{12} \]