Answer:
The rule for the linear function will be:
[tex]f(x) = -2x+5[/tex]
Step-by-step 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
[tex]\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\left(x_1,\:y_1\right)=\left(1,\:3\right),\:\left(x_2,\:y_2\right)=\left(2,\:1\right)[/tex]
[tex]m=\frac{1-3}{2-1}[/tex]
[tex]m=-2[/tex]
Using the point-slope form of the line equation
[tex]y-y_1=m\left(x-x_1\right)[/tex]
where m is the slope of the line and (x₁, y₁) is the point
substituting the values m = -2 and the point (1, 3)
[tex]y-y_1=m\left(x-x_1\right)[/tex]
[tex]y - 3 = -2 (x-1)[/tex]
[tex]y-3 = -2x+2[/tex]
adding 3 to both sides
[tex]y-3+3 = -2x+2+3[/tex]
[tex]y = -2x+5[/tex]
or
[tex]f(x) = -2x+5[/tex]
Therefore, the rule for the linear function will be:
[tex]f(x) = -2x+5[/tex]
