Question

Let f(x) = 5x + 12. Find f^−1(x).

Answer 1

`f^(-1)(x)=(x-12)/(5)`

Explanation:

Given:

`f(x)=5x+12`

Replace f(x) with y:

`y=5x+12`

Switch variables and solve for y:

`x=5y+12`

`x-12=5y`

`(x-12)/(5)=y`

Simplify:

`(1)/(5)x-(12)/(5)=y`

Replace y with f^-1(x):

`f^(-1)(x)=(1)/(5)x-(12)/(5)`

Therefore, the inverse function for
`f(x)=5x+12` is
`f^(-1)(x)=(1)/(5)x-(12)/(5)`.

Notice how in the graph attached of the inverse functions that they are symmetric about the line y=x. This is a crucial characteristic of inverse functions.

Answer 2

`\huge\boxed{f^(-1)(x)=(1)/(5)x-3}`

Explanation:

`f(x)=5x+15\to y=5x+15\\\\\text{exchange x to y and vice versa}\\\\x=5y+15\\\\\text{solve for}\ y\\\\5y+15=x\qquad|\text{subtract 15 from both sides}\\\\5y=x-15\qquad|\text{divide both sides by 5}\\\\y=(1)/(5)x-3`