` f ( x) = (3)/(2) x + b `
In the function above , b is a constant . If ` f (6) = 7`, what is the value of ` f ( -2 ) ` ?

Options

A. ` - 5 `
B. ` -2 `
C. ` 1 `
D. ` 7`

Explanation

To solve the problem step by step, we start with the function given:

\[ f(x) = \frac{3}{2}x + b \]

where \( b \) is a constant. We know that:

\[ f(6) = 7 \]

### Step 1: Substitute \( x = 6 \) into the function

We substitute \( x = 6 \) into the function to find \( b \):

\[ f(6) = \frac{3}{2}(6) + b \]

### Step 2: Calculate \( f(6) \)

Calculating \( \frac{3}{2}(6) \):

\[ \frac{3}{2}(6) = \frac{18}{2} = 9 \]

So, we have:

\[ f(6) = 9 + b \]

### Step 3: Set the equation equal to 7

Since we know that \( f(6) = 7 \), we can set up the equation:

\[ 9 + b = 7 \]

### Step 4: Solve for \( b \)

To find \( b \), we subtract 9 from both sides:

\[ b = 7 - 9 \]
\[ b = -2 \]

### Step 5: Rewrite the function with the value of \( b \)

Now that we have \( b \), we can rewrite the function:

\[ f(x) = \frac{3}{2}x - 2 \]

### Step 6: Find \( f(-2) \)

Next, we need to find \( f(-2) \). We substitute \( x = -2 \) into the function:

\[ f(-2) = \frac{3}{2}(-2) - 2 \]

### Step 7: Calculate \( f(-2) \)

Calculating \( \frac{3}{2}(-2) \):

\[ \frac{3}{2}(-2) = -3 \]

So we have:

\[ f(-2) = -3 - 2 \]
\[ f(-2) = -5 \]

### Final Answer

Thus, the value of \( f(-2) \) is:

\[ \boxed{-5} \]