If `f(x)=sqrt(x^(2)-1) and g(x)=(10)/(x+2)`, then `g(f(3))`=

Options

A. 0.2
B. 1.7
C. 2.1
D. 3.5

Explanation

To solve the problem step by step, we need to find the value of \( g(f(3)) \) given the functions \( f(x) = \sqrt{x^2 - 1} \) and \( g(x) = \frac{10}{x + 2} \).

### Step 1: Calculate \( f(3) \)
We start by substituting \( x = 3 \) into the function \( f(x) \).

\[ f(3) = \sqrt{3^2 - 1} \]

Calculating \( 3^2 - 1 \):

\[ 3^2 = 9 \quad \Rightarrow \quad 9 - 1 = 8 \]

Now, substituting back into the function:

\[ f(3) = \sqrt{8} \]

### Step 2: Calculate \( g(f(3)) \)
Now that we have \( f(3) = \sqrt{8} \), we need to find \( g(f(3)) \) which is \( g(\sqrt{8}) \).

Substituting \( \sqrt{8} \) into the function \( g(x) \):

\[ g(\sqrt{8}) = \frac{10}{\sqrt{8} + 2} \]

### Step 3: Simplify \( g(\sqrt{8}) \)
First, we simplify \( \sqrt{8} \):

\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]

Now substituting this back into \( g \):

\[ g(\sqrt{8}) = \frac{10}{2\sqrt{2} + 2} \]

Factoring out a 2 from the denominator:

\[ g(\sqrt{8}) = \frac{10}{2(\sqrt{2} + 1)} = \frac{5}{\sqrt{2} + 1} \]

### Step 4: Rationalize the Denominator
To simplify \( \frac{5}{\sqrt{2} + 1} \), we multiply the numerator and the denominator by \( \sqrt{2} - 1 \):

\[ g(\sqrt{8}) = \frac{5(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{5(\sqrt{2} - 1)}{2 - 1} = 5(\sqrt{2} - 1) \]

### Step 5: Approximate the Value
Now we can approximate \( 5(\sqrt{2} - 1) \). We know \( \sqrt{2} \approx 1.414 \):

\[ \sqrt{2} - 1 \approx 1.414 - 1 = 0.414 \]

Thus,

\[ 5(\sqrt{2} - 1) \approx 5 \times 0.414 \approx 2.07 \]

### Final Answer
Therefore, the value of \( g(f(3)) \) is approximately \( 2.07 \).