```markdown
**Examples**

- **Negative logarithm** \( f(x) = -\log x \)

  \[
  f^*(y) = \sup_{x > 0} (xy + \log x)
  \]

  \[
  = 
  \begin{cases} 
  -1 - \log(-y) & \text{if } y < 0 \\ 
  \infty & \text{otherwise} 
  \end{cases}
  \]

- **Strictly convex quadratic** \( f(x) = (1/2)x^T Q x \) with \( Q \in S_{++}^n \)

  \[
  f^*(y) = \sup_x (y^T x - (1/2)x^T Q x)
  \]

  \[
  = \frac{1}{2} y^T Q^{-1} y
  \]

*Source: A. d’Aspremont. Convex Optimization M2.*
```
