Question

Linear function f with the values f(-1)=8 and f(5)=6

Answer 1

The linear function is:

`f(x) = -(1)/(3)x-(23)/(3)\\`

Explanation:

Given

f(-1)=8 and f(5)=6

We can extract two pairs of input-output from the values given

From f(-1)=8,

(x1,y1) = (-1,8)

and

From f(5)=6

(x2,y2) = (5,6)

The linear function is given by:

`y = mx+b`

Here m is the slope which is calculated by the formula

`m =(y_2-y_1)/(x_2-x_1)`

Putting values

`m = (6-8)/(5+1)\\m = (-2)/(6)\\m = -(1)/(3)`

Putting in the equation

`y = -(1)/(3)x+b`

Putting the pair of input-output equation (5,6)

`6 = -(1)/(3)(5) +b\\6 = -(5)/(3)+b\\b = 6+(5)/(3)\\b = (18+5)/(3)\\b = (23)/(3)`

Putting the value of b

`y = -(1)/(3)x-(23)/(3)\\f(x) = -(1)/(3)x-(23)/(3)\\`

Hence,

The linear function is:

`f(x) = -(1)/(3)x-(23)/(3)\\`