If `f(x)=t^(2)+(3)/(2)t`, then `f(q-1)=`

Options

A. `q^(2)-(3)/(2)`
B. `q^(2)-(1)/(2)q`
C. `q^(2)+(1)/(2)q+(1)/(2)`
D. `q^(2)-(1)/(2)q-(1)/(2)`

Explanation

To solve the problem, we need to evaluate the function \( f(x) = t^2 + \frac{3}{2}t \) at \( t = q - 1 \).

### Step-by-Step Solution:

1. **Substitute \( t \) with \( q - 1 \)**:
\[ f(q - 1) = (q - 1)^2 + \frac{3}{2}(q - 1) \]

2. **Expand \( (q - 1)^2 \)**:
\[ (q - 1)^2 = q^2 - 2q + 1 \]

3. **Expand \( \frac{3}{2}(q - 1) \)**:
\[ \frac{3}{2}(q - 1) = \frac{3}{2}q - \frac{3}{2} \]

4. **Combine the results**:
\[ f(q - 1) = (q^2 - 2q + 1) + \left(\frac{3}{2}q - \frac{3}{2}\right) \]

5. **Combine like terms**:
- Combine \( -2q \) and \( \frac{3}{2}q \):
\[ -2q + \frac{3}{2}q = -\frac{4}{2}q + \frac{3}{2}q = -\frac{1}{2}q \]
- Combine the constant terms \( 1 \) and \( -\frac{3}{2} \):
\[ 1 - \frac{3}{2} = \frac{2}{2} - \frac{3}{2} = -\frac{1}{2} \]

6. **Final expression**:
\[ f(q - 1) = q^2 - \frac{1}{2}q - \frac{1}{2} \]

### Conclusion:
Thus, the value of \( f(q - 1) \) is:
\[ f(q - 1) = q^2 - \frac{1}{2}q - \frac{1}{2} \]