If f(x) is a linear function and f(2) = 1 and f(4) = -2 , then f(x) =

Options

A. `- (3)/(2) x + 4`
B. `(3)/(2) x - 2`
C. `- (3)/(2) x + 2`
D. `(3)/(2) x - 4`

Explanation

To find the linear function \( f(x) \) given the conditions \( f(2) = 1 \) and \( f(4) = -2 \), we can follow these steps:

### Step 1: Write the general form of a linear function
A linear function can be expressed in the form:
\[ f(x) = ax + b \]
where \( a \) and \( b \) are constants.

### Step 2: Set up equations using the given points
Using the information provided:
1. From \( f(2) = 1 \):
\[ 1 = 2a + b \quad \text{(Equation 1)} \]
2. From \( f(4) = -2 \):
\[ -2 = 4a + b \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
We have the following system of equations:
1. \( 2a + b = 1 \) (Equation 1)
2. \( 4a + b = -2 \) (Equation 2)

To eliminate \( b \), we can subtract Equation 1 from Equation 2:
\[ (4a + b) - (2a + b) = -2 - 1 \]
This simplifies to:
\[ 2a = -3 \]
Thus, we find:
\[ a = -\frac{3}{2} \]

### Step 4: Substitute \( a \) back to find \( b \)
Now, substitute \( a \) back into Equation 1 to find \( b \):
\[ 2\left(-\frac{3}{2}\right) + b = 1 \]
This simplifies to:
\[ -3 + b = 1 \]
So,
\[ b = 1 + 3 = 4 \]

### Step 5: Write the final function
Now that we have both \( a \) and \( b \), we can write the function:
\[ f(x) = -\frac{3}{2}x + 4 \]

### Final Answer
Thus, the linear function is:
\[ f(x) = -\frac{3}{2}x + 4 \]

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