How do you find the inverse of a function including its domain and range?

Question

The inverse function is one the will take the output of f(x) as an input and give you back x as its output.

If we have,

f(x) = (x - 2)²

With a domain = [2,∞)

The inverse of f(x) is,

f(f(x)) = x

To get the latter expression we just have to use some algebra to get x alone.

(x - 2)² = f(x)

x - 2 = √f(x)

x = √f(x) + 2

f(f(x)) = √f(x) + 2

so the inverse function of f(x) would be,

f(f(x)) = √f(x) + 2

Let's say x was equal to 3, then,

f(3) = (3 - 2)²

f(3) = 1

and the inverse function would be,

f(f(3)) = √f(3) + 2

f(f(3)) = √1 + 2

f(f(3)) = 1 + 2

f(f(3)) = 3

You can see that in the first function we substituted 3 for x and got 1 as the output and in the second function we substituted 1 for f(x) and got back 3.

The domain is all of the x (input) values that are defined for the function. The inverse function has x as its output, so the domain of the original function becomes the range, or all the possible output values, of the inverse function.

The domain of f(x) = [2,∞)

The range of f(f(x)) = [2,∞)

The domain of f(f(x)) is the range of f(x) so first, we need to figure out the minimum and maximum y values possible for f(x)

The smallest x can be is 2,

f(2) = (2 - 2)²

f(2) = 0

and the largest number is infinity,

f(∞) = (∞ - 2)²

f(∞) = ∞

Since f(x) has a range of [0,∞), f(f(x)) has the domain [0,∞)

Therefore the inverse function of f(x) = (x - 2)² on the domain = [2,∞) is,

f(f(x)) = √f(x) + 2

Domain = [0,∞)

Range = [2,∞)

Answer 1

How do you find the inverse of a function including its domain and range?

Let f(x) = 14 − x Calculate f^-1(x) = ??

Detailed Solution:

To find any inverse function, follow the following four steps:

1) Replace f(x) by y

f(x) = 14 – x ==> y = 14 – x

2) Interchange x and y

y = 14 – x ==> x = 14 – y

3) Solve for y

x = 14 – y ==> y = 14 – x

4) Replace the new y by f^-1(x) which is the inverse function solution

y = 14 – x ==> f^-1(x) = 14 – x <== Final Solution

Known Facts about Inverse Functions:

* Domain of f(x) is the same as Range f^-1(x)

* Range of f(x) is the same as Domain f^-1(x)

* Domain of f^-1(x) is the same as Range f(x)

* Range of f^-1(x) is the same as Domain f(x)

Now, Check for f(x) = 14 − x and f^-1(x) = 14 – x

1) (f o f^-1)(x) = x ==> f(f^-1(x)) = x ==> f(14 – x) = x ==> 14 – (14 − x) = x ==> 14 – 14 + x = x ==> x = x

2) (f^-1 o f)(x) = x ==> f^-1(f(x)) = x ==> f^-1(14 – x) = x ==> 14 – (14 − x) = x ==> 14 – 14 + x = x ==> x = x