Question

The function y = f (x) is linear.

Given f-¹ (0) = 3 and f (1) = - 4, find the formula of
f (x) and f-¹ (x)​

Answer 1

`f(x)=2x-6`

`f^(-1)(x)=(x+6)/(2)`

Explanation:

Using slope-intercept form of linear function:
`y = mx + b`

`\implies f(x)=mx+b`

`\textsf{if} \ \ f(1) = -4`

`\implies m + b=-4`

`\implies b = -4-m`

Find inverse of slope-intercept form:

swap x and y:
`x = my + b`

Make y the subject:

`\implies x - b = my`

`\implies y = (x-b)/(m)`

`\implies f^(-1)(x) = (x-b)/(m)`

`\textsf{if} \ \ f^(-1)(0) = 3`

`\implies (0-b)/(m)=3`

`\implies b=-3m`

`\textsf{equation 1:} \ \ b = -4-m`

`\textsf{equation 2:} \ \ b=-3m`

Equate the equations and solve for
`m`:

`\implies b=b`

`\implies -4-m=-3m`

`\implies -4=-2m`

`\implies m=2`

Substitute found value for
`m` into one of the equations and solve for
`b`:

`b = -4-2=-6`

Substitute found values of
`m` and
`b` into equations for
`f(x)` and
`f^(-1)(x)`:

`f(x)=2x-6`

`f^(-1)(x)=(x+6)/(2)`