A function is considered proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant. This means that for any inputs \( x \) and \( y \), the ratio \( \frac{y}{x} \) should be constant.
Let's analyze the two functions you've provided:
Function 1:
\[ \begin{align*} x & : 1 \quad \quad y & : 2 \quad \Rightarrow \quad \frac{y}{x} = \frac{2}{1} = 2 \ x & : 2 \quad \quad y & : 4 \quad \Rightarrow \quad \frac{y}{x} = \frac{4}{2} = 2 \ x & : 3 \quad \quad y & : 6 \quad \Rightarrow \quad \frac{y}{x} = \frac{6}{3} = 2 \ \end{align*} \]
In Function 1, the ratio \( \frac{y}{x} \) is consistently 2, indicating it is proportional.
Function 2:
\[ \begin{align*} x & : 1 \quad \quad y & : 2 \quad \Rightarrow \quad \frac{y}{x} = \frac{2}{1} = 2 \ x & : 2 \quad \quad y & : 3 \quad \Rightarrow \quad \frac{y}{x} = \frac{3}{2} = 1.5 \ x & : 3 \quad \quad y & : 4 \quad \Rightarrow \quad \frac{y}{x} = \frac{4}{3} \approx 1.33 \ \end{align*} \]
In Function 2, the ratios \( \frac{y}{x} \) are not constant (2, 1.5, and approximately 1.33), indicating it is not proportional.
Conclusion: Function 1 is proportional.