Answer
$f(x)=x-\dfrac{3}{4}$
Work Step by Step
Using $y-y_1=\dfrac{y_1-y_2}{x_1-x_2}(x-x_1)$ or the Two-Point Form of linear equations, then the equation of the line passing through $
\left( \dfrac{1}{2},-\dfrac{1}{4} \right) \text{ and } \left( \dfrac{3}{2},\dfrac{3}{4} \right)
,$ is
\begin{array}{l}\require{cancel}
y-\left( -\dfrac{1}{4} \right)=\dfrac{-\dfrac{1}{4}-\dfrac{3}{4}}{\dfrac{1}{2}-\dfrac{3}{2}}\left(x-\dfrac{1}{2} \right)
\\\\
y+\dfrac{1}{4}=\dfrac{-\dfrac{4}{4}}{-\dfrac{2}{2}}\left(x-\dfrac{1}{2} \right)
\\\\
y+\dfrac{1}{4}=\dfrac{-1}{-1}\left(x-\dfrac{1}{2} \right)
\\\\
y+\dfrac{1}{4}=x-\dfrac{1}{2}
\\\\
y=x-\dfrac{1}{2}-\dfrac{1}{4}
\\\\
y=x-\dfrac{2}{4}-\dfrac{1}{4}
\\\\
y=x-\dfrac{3}{4}
.\end{array}
In function notation, this is equivalent to $
f(x)=x-\dfrac{3}{4}
.$