Question

How do you find `f^-1(x)` given `f(x)=2x+7`?

Answer 1

`f^-1(x)=1/2(y-7)`

Explanation:

Given:
`f(x)=2x+7`

Let y=f(x)

`y=2x+7`

Expressing x in terms of y gives us the inverse of x

`y-7=2x`

`2x=y-7`

`x=1/2(y-7)`

Thus,

`f^-1(x)=1/2(y-7)`

Answer 2

The `f^{-1}` notation indicates you need to find the inverse of the function

Explanation:

There are a couple of ways to look at function inverses. An inverse of anything allows you to 'undo' whatever you started with. So, if you tie your shoe, it's not there forever - you can always untie it.

We have many inverse functions in math, such as square root is the inverse of squaring a number, etc.

Finding the inverse also reflects the graph across the line y = x.

There are 3 steps to finding an inverse:

1) change notation `f(x) =` to y =

So, y = 2x + 7

2) Exchange the x & y variables. Note this is what accomplishes that reflection across the line y = x

So, x = 2y + 7

3) Since x is the dependent variable and y is the independent variable and it is always a zillion times easier to solve a problem in y = form, solve the equation for y.

First subtract 7 from both sides

x - 7 = 2y

Then divide by 2

`y = {x-7}/2`