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What is the Inverse of a function?

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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?

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"Mathy" Answer
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 equivalen to what kind of transformation question
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) }
applet Explore the relationship of the domain and range more, with our interactive function applet

Is the inverse of a function always a function?

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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?
Answer
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
Practice Problems I


Problem 1)
f(x) = { (1,2) , (3,4) , (5,6) }
What is the inverse of this function?
Answer
Load More Problems Like This
Solving algebraically for the Inverse

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

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


Problem 1) What is the inverse of the function f(x) = x + 22?
Step 1) replace f(x) with y question y = x + 22
Step2) switch x and y question x = y + 22
switch x and y
Step 3) solve for new 'y' question work
Step 4) replace 'y' with f-1(x) question f-1(x) = x - 22


Problem 2) What is the inverse of the function f(x) = 2x ?
Step 1) replace f(x) with y question y = 2x
Step2) switch x and y question x = 2y
switch x and y
Step 3) solve for new 'y' question
Step 4) replace 'y' with f-1(x) question


Problem 3) What is the inverse of the function $$ f(x) = \frac{1}{2}x + 3 $$ ?
Step 1) replace f(x) with y question
Step2) switch x and y question
Step 3) solve for new 'y' question
Step 4) replace 'y' with f-1(x) question


Problem 4) What is the inverse of the function $$ f(x) = x^2 + 3 $$ ?
Step 1) replace f(x) with y question
Step2) switch x and y question
Step 3) solve for new 'y' question
Step 4) replace 'y' with f-1(x) question


Problem 5) What is the inverse of the function $$ f(x) = 5 \sqrt{ x } $$ ?
Step 1) replace f(x) with y question
Step2) switch x and y question
Step 3) solve for new 'y' question
Step 4) replace 'y' with f-1(x) question