If f(x+2)=3x+11 and g(f(x))=2x, find the value of g(5).

Options

A. 0
B. 1
C. 3
D. 5

Answer

To solve the problem, we need to find the value of \( g(5) \) given the functions \( f(x+2) = 3x + 11 \) and \( g(f(x)) = 2x \).

### Step 1: Find the function \( f(x) \)

We start with the equation:
\[ f(x+2) = 3x + 11 \]
To express \( f(x) \), we can substitute \( x \) with \( x-2 \):
\[ f((x-2)+2) = 3(x-2) + 11 \]
This simplifies to:
\[ f(x) = 3(x-2) + 11 = 3x - 6 + 11 = 3x + 5 \]

### Step 2: Substitute \( f(x) \) into \( g(f(x)) \)

Now we know:
\[ f(x) = 3x + 5 \]
We substitute this into the equation \( g(f(x)) = 2x \):
\[ g(3x + 5) = 2x \]

### Step 3: Find \( x \) such that \( f(x) = 5 \)

We need to find the value of \( x \) for which \( f(x) = 5 \):
\[ 3x + 5 = 5 \]
Subtracting 5 from both sides gives:
\[ 3x = 0 \]
Dividing both sides by 3 results in:
\[ x = 0 \]

### Step 4: Calculate \( g(5) \)

Now that we have \( x = 0 \), we can substitute this back into \( g(f(x)) \):
\[ g(5) = g(3(0) + 5) = g(5) = 2(0) = 0 \]

### Final Answer

Thus, the value of \( g(5) \) is:
\[ \boxed{0} \]

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