Inverse of a Function

What is the Inverse of a function?

A conceptual analogy

If the function itself is considered a "DO" action, then the inverse is the "UNDO"

inverse of a function flow chart

What about the domain and Range?

Definition: The inverse of a function is when the domain and the range trade places. All elements of the domain become the range, and all elements of the range become the domain.


Therefore, the inverse of a function is equivalent to what kind of transformation

Reflection over the line y = x

Inverse of a function domain and range switch domain and range function Another Example
Original function f(x) Inverse of function or f-1(x)

{ (0,3 ) , (1,4) , (2, 5) }

{ (3, 0 ) , (4,1) , (5, 2) }

Explore the relationship of the domain and range more, with our interactive function applet

Is the inverse of a function always a function?

To answer that question let's look at the function in Diagram 1. In that graph, you can see
the original function and its inverse

Is the inverse of that function, also a function?

No

As you can see, the inverse of that function does not pass the vertical line test and, therefore , is not a function.

Diagram I Picture of inverse of a function

Conclusion: The inverse of a function is not necessarily a function.

So...when is the inverse of a function also a function?

f-1(x), the inverse, is itself a function only when f(x), the original function, is a one-to-one function.

Remember: 1 to 1 functions must pass the horizontal line test!

Example 1

In this case, f(x) is a function, but f-1(x) is not a function.

not a one to one function

Function #1, f(x) is not a one to one function because it does not pass the horizontal line test.

not a one to one function

f-1(x) is not a function

Example 2

In this case, f(x) is a function and f-1(x) is also a function.

example 2 inverse of function

This function passes the horizontal line test so, its inverse will also be a function.

example 2 inverse

Practice Problems Part I

Problem 1

f(x) = { (1,2) , (3,4) , (5,6) } What is the inverse of this function?

Remember: $$ f^{-1} (x) $$ means 'the inverse of f(x) '

$$ f^{-1} (x) $$ = { (2,1) , (4,3) , (6,5) }

Problem 2

f(x) = { (33,14) , (23,15) , (11,12), (13, ,14 )} What is the inverse of this function?

Remember: $$ f^{-1} (x) $$ means 'the inverse of f(x) '

$$ f^{-1} (x) $$ = { (14,33) , (15, 23) , (12,11) , (14, 13) }}

Problem 3

f(x) = { (-1, 12) , (13, 114) , (15,61), ( 1, 12)} What is the inverse of this function?

Remember: $$ f^{-1} (x) $$ means 'the inverse of f(x) '

$$ f^{-1} (x) $$ = { (12, -1) , (114, 13) , (61,15) , ( 12,1) }

Problem 4

What is the inverse of y = 3

y = 3 is a horizontal line. In a horizontal line all elements of the range are 3.

So, it looks something $$ f^{-1} (x) $$ ={ ( -1,3 ) , (0,3) , (1,3)}

Therefore the inverse of this function will be whatever line has 3 for all elements in its domain.

Therefore the inverse of y = 3 is the line x = 3

Solving Algebraically for the Inverse

What is the inverse of f(x) = x + 1?

Just like in our prior examples, we need to switch the domain and range. In an equation , the domain is represented by the   x variable and the range by the   y variable

steps for solving for inverse of function
To Summarize

f(x): took an element from the domain and added 1 to arrive at the corresponding element in the range.

f-1(x) : took an element from the domain and subtracted 1 to arrive at the corresponding element in the range.

Practice Problems Part II

Problem 1

What is the inverse of the function f(x) = x + 22?

replace f(x) with y

y = x + 22

switch x and y

x = y + 22

solve for new 'y'
work
replace 'y' with f-1(x)
f-1(x) = x - 22
Problem 2

What is the inverse of the function f(x) = 2x?

replace f(x) with y

y = 2x

switch x and y

x = 2y

switch x and y
solve for new 'y'

$$ \frac{1}{2} x = \frac{1}{2} \cdot 2y \\ \frac{1}{2} x = y $$

replace 'y' with f-1(x)

$$ f^{-1 } (x) =\frac{1}{2} x $$

Problem 3

What is the inverse of the function $$ f(x) = \frac{1}{2}x + 3 $$?

replace f(x) with y

$$ y = \frac{1}{2}x + 3 $$

switch x and y

$$ x = \frac{1}{2}y + 3 $$

switch x and y
solve for new 'y'

$$ \color{Red}{2 \cdot } x = \color{Red}{2 \cdot } \big ( \frac{1}{2}y + 3 \big) \\ 2 \cdot x = \frac{2}{2}\cdot y + 2 \cdot 3 \\ 2x = y + 6 \\ 2x -6 = y $$

replace 'y' with f-1(x)

$$ f^{-1 } (x) = 2x -6 $$

Problem 4

What is the inverse of the function $$ f(x) = x^2 + 3 $$?

replace f(x) with y

$$ y = x^2 + 3 $$

switch x and y

$$ x = y^2 + 3 $$

switch x and y
solve for new 'y'

$$ x \color{Red}{ -3} = y^2 + 3 \color{Red}{ -3} \\ x -3 = y^2 \\ \sqrt{x-3} = \sqrt{y^2} \\ \sqrt{x-3} = y $$

replace 'y' with f-1(x)

$$ f^{-1 } (x) = \sqrt{x-3} $$

Problem 5

What is the inverse of the function $$ f(x) = 5 \sqrt{ x } $$ ?

replace f(x) with y

$$ y = 5 \sqrt{ x } $$

switch x and y

$$ x = 5 \sqrt{ y } $$

switch x and y
solve for new 'y'

$$ x{ \color{Red}{^2}} = \big( 5 \sqrt{ y } \big) { \color{Red}{^2}} \\ x^2 = 5{ \color{Red}{^2}} \cdot ( \sqrt{ y } ){ \color{Red}{^2}} \\ x^2 = 25y \\ \frac{1}{25} \cdot x^2 = \frac{1}{25} \cdot 25y \\ \frac{1}{25} \cdot x^2 = y $$

replace 'y' with f-1(x)

$$ f^{-1 } (x) = \frac{1}{25} \cdot x^2 $$

Next Lesson Inverse of Function Applet