Question

Hi! I'm starting my Algebra Coursework!

I went through the lessons already, but I'm still not quite understanding how to find the inverse of a function! In this case, I need to find the inverse of f(x).

I'll write one of the practice problems the program gave me, and if you could help explain (detailed) how to solve it, that would be great! :D

*I am already aware that f(x) is the same as y. But after that, I'm so lost!*

Find the inverse of f(x).
f(x) = 8x + 7

Thank you! I just want to be prepared for the upcoming test! ^^

Answer 1

`\huge\boxed{\sf f^(-1)(x)=(x-7)/(8)}`

Explanation:

Given function:

f(x) = 8x + 7

• Put f(x) = y.

y = 8x + 7

• Exchange x and y.

x = 8y + 7

• Now, solve for y.

x = 8y + 7

• Subtract 7 from both sides.

x - 7 = 8y

• Divide both sides by 8.

`\displaystyle (x-7)/(8) = y`

OR

`\displaystyle y = (x-7)/(8)`

• Now, put y = f⁻¹(x).

So,

`\displaystyle f^(-1)(x)=(x-7)/(8) \\\\\rule[225]{225}{2}`

Answer 2

`f^(-1)(x) = (x-7)/(8)`

Explanation:

Before we start looking at how to find the inverse of a function, we first need to establish what it is.

A function is a set of operations done onto an input (x) to produce an output (y). A function's inverse is a set of operations that, when done onto the original function's output (y), returns the input (x).

A simple example of this would be:

f(x) = x + 3 ... This function adds 3 to the input to produce its output.

Here are some inputs and their corresponding outputs for this function:

In | Out

0 | 3

1 | 4

2 | 5

So the inverse of that function would produce the following inputs and outputs:

In | Out

3 | 0

4 | 1

5 | 2

In this example, we can deduce that the inverse of the function f(x) = x + 3 is:

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

because subtraction is the opposite operation of addition.

Now that we have established what an inverse function is, here are the 2 main methods for finding the inverse of a function:

1.

(we just did this) Listing out the operations done onto x and inverting the order of their opposites ...

Given the function: f(x) = 8x + 7

We can see that the operations onto x are:

1. multiply by 8
2. add 7

Now, to create the inverse function, we will invert the order of each step's opposite operation:

1. subtract 7
2. divide by 8

So, the inverse of the given function is:

`f^(-1)(x) = (x-7)/(8)`

or in text form:
`f^(-1)`(x) = (x - 7) / 8

2.

Swapping f(x) with x and x with
`f^(-1)`(x), then solving for
`f^(-1)`(x) ...

Given the function: f(x) = 8x + 7

↓ swapping f(x) with x and x with
`f^(-1)`(x)

`x = 8 \cdot f^(-1)(x) + 7`

↓ subtracting 7 from both sides

`x - 7 = 8 \cdot f^(-1)(x)`

↓ dividing both sides by 8

`(x - 7)/(8) = f^(-1)(x)`