Function f is defined by `f(x)=(3)/(2)x+c`. If `f(6)=1`, what is the value of `f(c)`?

Options

A. `-20`
B. `-8`
C. `4`
D. `12`

Answer

To solve the problem step by step, we start with the function definition and the given condition.

### Step 1: Write down the function and the condition
The function is given by:
\[ f(x) = \frac{3}{2}x + c \]
We also know that:
\[ f(6) = 1 \]

### Step 2: Substitute \( x = 6 \) into the function
We substitute \( x = 6 \) into the function to find \( c \):
\[ f(6) = \frac{3}{2}(6) + c \]

### Step 3: Calculate \( f(6) \)
Calculating \( \frac{3}{2}(6) \):
\[ \frac{3}{2} \times 6 = 9 \]
So we have:
\[ f(6) = 9 + c \]

### Step 4: Set up the equation
Since we know that \( f(6) = 1 \), we can set up the equation:
\[ 9 + c = 1 \]

### Step 5: Solve for \( c \)
To find \( c \), we subtract 9 from both sides:
\[ c = 1 - 9 \]
\[ c = -8 \]

### Step 6: Substitute \( c \) back into the function
Now we need to find \( f(c) \), which is \( f(-8) \):
\[ f(-8) = \frac{3}{2}(-8) + c \]

### Step 7: Substitute \( c = -8 \) into the function
Now we substitute \( c = -8 \) into the function:
\[ f(-8) = \frac{3}{2}(-8) - 8 \]

### Step 8: Calculate \( f(-8) \)
Calculating \( \frac{3}{2}(-8) \):
\[ \frac{3}{2} \times -8 = -12 \]
So we have:
\[ f(-8) = -12 - 8 \]
\[ f(-8) = -20 \]

### Final Answer
Thus, the value of \( f(c) \) is:
\[ \boxed{-20} \]