The equation for the function is [tex]y=x+\frac{1}{2}[/tex]
Explanation:
The equation for the function that passes through the points [tex]\left(1, \frac{3}{2}\right)[/tex] and [tex]\left(\frac{3}{2}, 2\right)[/tex]
The equation of the line that passes through the points [tex]\left(x_{1}, y_{1}\right)[/tex] and [tex]\left(x_{2}, y_{2}\right)[/tex] is given by
[tex]y-y_{1}=m\left(x-x_{1}\right)[/tex]
where m is the slope and it can be determined using the formula,
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
Substituting the values of x and y in the slope formula, we have,
[tex]m=\left(\frac{2-\frac{3}{2}}{\frac{3}{2}-1}\right)[/tex]
Simplifying, we get,
[tex]\begin{aligned}&m=\frac{\left(\frac{1}{2}\right)}{\left(\frac{1}{2}\right)}\\&m=1\end{aligned}[/tex]
Thus, the slope is m=1.
Substituting the values in the equation [tex]y-y_{1}=m\left(x-x_{1}\right)[/tex], we have,
[tex]y-\frac{3}{2} =1(x-1)[/tex]
Multiplying the terms within the bracket,
[tex]y-\frac{3}{2} =x-1[/tex]
Adding [tex]\frac{3}{2}[/tex] on both sides of the equation,
[tex]y=x+\frac{1}{2}[/tex]
Thus, the equation for the function is [tex]y=x+\frac{1}{2}[/tex]