Question

What is the inverse of the function `f(x) = -3x + 3`?

Answer 1

`f^-1(x) = (3-x)/3 quad` Step by step working is shown below.

Explanation:

To find inverse function follow the following steps

Step 1: Swap `x` and `y`
Step 2: Solve for `y`
Step 3: The result you got is the inverse function. Write in proper notation.

Let us see this with the given problem.

Let `y=f(x) = -3x+3`

`y=-3x+3`

Step 1: Swap `x` and `y`
`x=-3y+3`

Step 2: Solve for `y`

`x-3 = -3y quad` On subtracting 3 from both sides.
`(x-3)/-3= y quad` Dividing by -3 on both sides isolates `y`

`(3-x)/3 = y`

Step 3: Writing it in proper notation

`f^-1(x) = (3-x)/3`

Answer 2

`bar(f(x)) = 1-x/3`
`color(white)("XXXXXXXXX")`where `bar(f(x))` denotes the inverse of `f(x)=-3x+3`

Explanation:

Let `bar(f(x))` be the inverse of `f(x)=-3x+3`

By definition of inverse
`color(white)("XXX")f(bar(f(x)))=color(blue)(x)`

But if `f(color(red)(x)) = -3color(red)(x)+3`
then
`color(white)("XXX")f(color(red)(bar(f(x))))=-3color(red)(bar(f(x)))+3`

Therefore
`color(white)("XXX")-3bar(f(x))+3=color(blue)(x)`

`color(white)("XXX")-3bar(f(x)) = x-3`

`color(white)("XXX")bar(f(x)) = (x-3)/(-3)`

`color(white)("XXX")bar(f(x))=1-x/3`