The function f and g are defined as `f(x) = 2x - 3` and g(x) = x + 3//2`. For what value of y is `f(y) = g(y- 3)` ?

Options

A. `1`
B. `3//2`
C. `2`
D. `3`

Answer

To solve the problem, we need to find the value of \( y \) for which \( f(y) = g(y - 3) \).

Given the functions:
- \( f(x) = 2x - 3 \)
- \( g(x) = \frac{x + 3}{2} \)

### Step 1: Write down the expressions for \( f(y) \) and \( g(y - 3) \)

1. **Calculate \( f(y) \)**:
\[ f(y) = 2y - 3 \]

2. **Calculate \( g(y - 3) \)**:
\[ g(y - 3) = \frac{(y - 3) + 3}{2} = \frac{y}{2} \]

### Step 2: Set the equations equal to each other

Now, we set \( f(y) \) equal to \( g(y - 3) \):
\[ 2y - 3 = \frac{y}{2} \]

### Step 3: Clear the fraction

To eliminate the fraction, multiply both sides of the equation by 2:
\[ 2(2y - 3) = y \]
This simplifies to:
\[ 4y - 6 = y \]

### Step 4: Rearrange the equation

Now, we rearrange the equation to isolate \( y \):
\[ 4y - y = 6 \]
This simplifies to:
\[ 3y = 6 \]

### Step 5: Solve for \( y \)

Now, divide both sides by 3:
\[ y = 2 \]

### Step 6: Verify the solution

To ensure our solution is correct, we can substitute \( y = 2 \) back into both functions:
1. Calculate \( f(2) \):
\[ f(2) = 2(2) - 3 = 4 - 3 = 1 \]

2. Calculate \( g(2 - 3) = g(-1) \):
\[ g(-1) = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \]

Since \( f(2) = g(-1) = 1 \), our solution is verified.

### Final Answer
The value of \( y \) for which \( f(y) = g(y - 3) \) is:
\[ \boxed{2} \]