If `f(x)=sqrt(2x+3)` and `g(x)=x^(2)+1`, then `f(g(2))=`

Options

A. 2.24
B. 3
C. 3.61
D. 6

Explanation

To solve the problem step by step, we will find the value of \( f(g(2)) \) using the functions provided.

### Step 1: Find \( g(2) \)

The function \( g(x) \) is defined as:
\[ g(x) = x^2 + 1 \]

Now, we need to calculate \( g(2) \):
\[ g(2) = 2^2 + 1 \]
Calculating \( 2^2 \):
\[ g(2) = 4 + 1 = 5 \]

### Step 2: Find \( f(g(2)) \)

Now that we have \( g(2) = 5 \), we need to find \( f(5) \).

The function \( f(x) \) is defined as:
\[ f(x) = \sqrt{2x + 3} \]

Now, we will calculate \( f(5) \):
\[ f(5) = \sqrt{2 \cdot 5 + 3} \]
Calculating \( 2 \cdot 5 \):
\[ f(5) = \sqrt{10 + 3} \]
Now, adding \( 10 + 3 \):
\[ f(5) = \sqrt{13} \]

### Step 3: Final Result

Thus, the value of \( f(g(2)) \) is:
\[ f(g(2)) = \sqrt{13} \]

### Step 4: Approximate Value (Optional)

If we want to approximate \( \sqrt{13} \), we can note that:
\[ 3^2 = 9 \quad \text{and} \quad 4^2 = 16 \]
This means \( \sqrt{13} \) is between 3 and 4. Using a calculator, we find:
\[ \sqrt{13} \approx 3.61 \]

### Summary of the Solution
The final answer is:
\[ f(g(2)) = \sqrt{13} \approx 3.61 \]