Answer:
The rule for the linear function will be:
![f(x) = -2x+5](https://img.qammunity.org/2022/formulas/mathematics/high-school/wjix0afhp80g7f0oc8ww7shgcmjuyfgzm8.png)
Explanation:
We know that linear function can be represented using the slope-intercept formula
y = mx+b
where m is the slope and b is the y-intercept
Given the function is linear
Taking two points from the table to determine the slope
![\mathrm{Slope}=(y_2-y_1)/(x_2-x_1)](https://img.qammunity.org/2022/formulas/mathematics/high-school/noa3dwrz4s6a4umc1ibrxg0crgl23zrf2o.png)
![\left(x_1,\:y_1\right)=\left(1,\:3\right),\:\left(x_2,\:y_2\right)=\left(2,\:1\right)](https://img.qammunity.org/2022/formulas/mathematics/high-school/dq8hpawj7l4nr5adlyvahssz8hfy3icea6.png)
![m=(1-3)/(2-1)](https://img.qammunity.org/2022/formulas/mathematics/high-school/krq76ubvh8eva9hrrvz7twqvhgh2651nyl.png)
![m=-2](https://img.qammunity.org/2022/formulas/mathematics/high-school/sqgur7tnwagqh99prj84y03u1yeo5x82v5.png)
Using the point-slope form of the line equation
![y-y_1=m\left(x-x_1\right)](https://img.qammunity.org/2022/formulas/mathematics/high-school/59wu8ly47al9vwq2ng2tgmsrx1lo1l4azh.png)
where m is the slope of the line and (x₁, y₁) is the point
substituting the values m = -2 and the point (1, 3)
![y-y_1=m\left(x-x_1\right)](https://img.qammunity.org/2022/formulas/mathematics/high-school/59wu8ly47al9vwq2ng2tgmsrx1lo1l4azh.png)
![y - 3 = -2 (x-1)](https://img.qammunity.org/2022/formulas/mathematics/high-school/984nixfxezcr8m59ro89dzh2dmw6pru77i.png)
![y-3 = -2x+2](https://img.qammunity.org/2022/formulas/mathematics/high-school/tye8xhhjx3ttzo4jaup0a7h43ajg37iwn6.png)
adding 3 to both sides
![y-3+3 = -2x+2+3](https://img.qammunity.org/2022/formulas/mathematics/high-school/geja8zew4je3620u8kpmq37251r7glgwy9.png)
![y = -2x+5](https://img.qammunity.org/2022/formulas/mathematics/high-school/dlzdb4f1y199hwu1shf24m323tte3zvmv1.png)
or
![f(x) = -2x+5](https://img.qammunity.org/2022/formulas/mathematics/high-school/wjix0afhp80g7f0oc8ww7shgcmjuyfgzm8.png)
Therefore, the rule for the linear function will be:
![f(x) = -2x+5](https://img.qammunity.org/2022/formulas/mathematics/high-school/wjix0afhp80g7f0oc8ww7shgcmjuyfgzm8.png)