A function has the form f(x)=ax+b, where a and b are constants. If `f(2)=1" and "f(-3)=11`, the function is defined by

Options

A. `f(x)=2x+5`
B. `f(x)=2x-5`
C. `f(x)=-2x+5`
D. `f(x)=-2x+5`

Answer

To solve the problem, we need to find the constants \( a \) and \( b \) in the linear function \( f(x) = ax + b \) given the conditions \( f(2) = 1 \) and \( f(-3) = 11 \).

### Step 1: Set up the equations based on the given conditions

From the first condition \( f(2) = 1 \):
\[ f(2) = a(2) + b = 1 \implies 2a + b = 1 \quad \text{(Equation 1)} \]

From the second condition \( f(-3) = 11 \):
\[ f(-3) = a(-3) + b = 11 \implies -3a + b = 11 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

Now we have a system of two equations:
1. \( 2a + b = 1 \) (Equation 1)
2. \( -3a + b = 11 \) (Equation 2)

We can eliminate \( b \) by subtracting Equation 1 from Equation 2:
\[ (-3a + b) - (2a + b) = 11 - 1 \]
This simplifies to:
\[ -3a - 2a = 10 \implies -5a = 10 \]
Dividing both sides by -5 gives:
\[ a = -2 \]

### Step 3: Substitute \( a \) back to find \( b \)

Now that we have \( a = -2 \), we can substitute this value back into Equation 1 to find \( b \):
\[ 2(-2) + b = 1 \]
This simplifies to:
\[ -4 + b = 1 \implies b = 1 + 4 = 5 \]

### Step 4: Write the final function

Now that we have both constants, \( a \) and \( b \):
\[ a = -2, \quad b = 5 \]
Thus, the function is:
\[ f(x) = -2x + 5 \]

### Final Answer:
The function defined by the given conditions is:
\[ f(x) = -2x + 5 \]