Inverse of a Function - Facts

Inverse Functions - What are they? - Video

Definition One-to-One

Definition One-to-One:

There are two conditions for a function to be one-to-one.

1 - If any two different input values are substituted into a function then there output values cannot be the same.

2 - If any two output values are the same, then their input values must be the same (i.e. the ordered pair is written twice).

Horizontal Line Test

Horizontal Line Test:

This test helps us determine if a function is one-to-one. It shows us if the inverse will be a function. If the horizontal line touches the function more than once, then there are two output values. Which means we find the inverse, those two output values will become input values, but we would have two different output values for the inverse.

Let's show that with some numbers. If one input is 4 and the output is 7 and another input is -5 and the output is 7, our inverse would flip flop those numbers. So we would have one input would be 7 and the output is 4 and the other input would 7, but the output would be -5. This scenario would follow the model on the bottom right in the picture of definition of one-to-one.

Theorem - Increasing or Decreasing Functions are One-to-One

Theorem - Increasing or Decreasing Functions are One-to-One:

This is stating that if a function is always increasing or decreasing then the function is one-to-one.

Let's draw a line that has a positive or negative slope. That line is always going to be increasing or decreasing. But, once we start adding some turns and/or extrema, there is some potential for repeat values like with a parabolic or absolute value function. 

Definition of Inverse Function

Definition of Inverse Function:

If we have two functions f and m, then m can only be the inverse of f if and only if y = f(x) and x = m(y). 

This means the input values, x, of f will be become the output values of m, y, and the output values of f, f(x), will become the input values, m(y), of m.

Theorem of Inverse Function

Theorem of Inverse Function:

If we have two functions f and m, then m can only be the inverse of f if and only if m(f(x)) = x for every x in D and f(m(y)) = y for every y in R. 

Condition 1 - We start with x. Then we substitute x, the input value, into f(x) and our result, the output value, is f(x) or y. Then that output value, f(x), is the input value of m(x). After substituting f(x) or y into function m(x), our output value will be m(f(x)).

Condition 2 - We start with y. Then we substitute y, the input value, into m(x) and our result, the output value, is m(y). Then that output value, m(y), is the input value of f(x). After substituting m(y) into function f(x), our output value will be f(m(y)).

This leads to what the domain and range of the function and its inverse.

Domain and Range of f and f -1 

Domain and Range of f and f1:

This tells us that the that range of f(x) is the domain of m(x) and the range of m(x) is the domain of f(x).

Guidelines for Finding the Inverse 

Guidelines for Finding the Inverse:

1 - Verify the function is one-to-one. Look at the domain. Is it increasing/decreasing throughout? Does it pass the horizontal line test?

2 - If yes. Then solve x in terms in y. 

3 - Verify if f(f-1(x)) and f-1(f(x)) = x. 

Live Worksheet

Teacher - Edpuzzle Link