3.2: OnetoOne Functions and Their Inverses
The statement "Pizza restaurants sell pizza" could be thought of as a function. It could be plotted on a graph, with different restaurants across the xaxis, and different foods the restaurant specializes in on the yaxis. Any time a pizza restaurant was input into the function, it would output "pizza" as the specialized food.
Is this pizza restaurant function a 1 to 1 function? How can we tell?
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 James Sousa: Animation: Inverse Function
Guidance
Consider the function \begin{align*}f(x)=x^3\end{align*}, and its inverse \begin{align*}f^{1} (x) = \sqrt[3]{x}\end{align*}.
The graphs of these functions are shown below:
The function f(x) = x^{3} is an example of a onetoone function, which is defined as follows:


The function y = x^{2}, however, is not onetoone. The graph of this function is shown below.
You may recall that you can identify a relation as a function if you can draw a vertical line anywhere through the graph, and the line touches only one point.
Notice then that if we draw a horizontal line through y = x^{2}, the line touches more than one point. That indicates that the inverse will not be a function, here is why: If we invert the function y = x^{2}, the result is a graph that is a reflection over the line y = x, effectively rotating the original 90deg. Since x and y have swapped, the new function fails the vertical line test.
The function y = x^{2} is therefore not a onetoone function. A function that is onetoone will be invertible.
You can determine an invertible function graphically by drawing a horizontal line through the graph of the function, if it touches more than one point, the function is not invertible.
Example A
Graph the function \begin{align*}f(x) = \frac{1} {3}x+2\end{align*}. Use a horizontal line test to verify that the function is invertible.
Solution: The graph below shows that this function is invertible. We can draw a horizontal line at any y value, and the line will only cross \begin{align*}f(x) = \frac{1} {3}x + 2\end{align*} once.
In sum, a onetoone function is invertible. That is, if we invert a onetoone function, its inverse is also a function. Now that we have established what it means for a function to be invertible, we will focus on the domain and range of inverse functions.
Example B
State the domain and range of the function and its inverse:
 Function: (1, 2), (2, 5), (3, 7)
Solution:
The inverse of this function is the set of points (2, 1), (5, 2), (7, 3)
 The domain of the function is {1, 2, 3}. This is also the range of the inverse.
 The range of the function is {2, 5, 7}. This is also the domain of the inverse.
The linear functions we examined previously, as well as f(x) = x^{3}, all had domain and range both equal to the set of all real numbers. Therefore the inverses also had domain and range equal to the set of all real numbers. Because the domain and range were the same for these functions, switching them maintained that relationship.
Also, as we found above, the function y = x^{2} is not onetoone, and hence it is not invertible. That is, if we invert it, the resulting relation is not a function. We can change this situation if we define the domain of the function in a more limited way. Let f(x) be a function defined as follows: f(x) = x^{2}, with domain limited to real numbers ≥ 0. Then the inverse of the function is the square root function: \begin{align*}f^{1} (x) = \sqrt{x}\end{align*}
Example C
Define the domain for the function f(x) = (x  2)^{2} so that f is invertible.
Solution:
The graph of this function is a parabola. We need to limit the domain to one side of the parabola. Conventionally in cases like these we choose the positive side; therefore, the domain is limited to real numbers ≥ 2.
Have you considered the question from the beginning of the lesson? "Pizza restaurants sell pizza" is a function. However, it is NOT a 1 : 1 function. In order to be 1 : 1, it must be invertible, giving something like: "pizza sellers are pizza restaurants", and that statement must also be a function. Since grocery stores sell pizza, and would therefore be among the outputs of the new function, but were not among the inputs of the original (which specified "pizza restaurants"), the functions are not invertible. 

Vocabulary
The inverse of a function is the relation obtained by interchanging the domain and range of a function.
A function is invertible if its inverse is a function.
A function is onetoone if every element of its domain is paired with exactly one element of its range.
Guided Practice
1) Is \begin{align*}g(x) = 3x  2\end{align*} a one to one function?
2) Use the horizontal line test to see if \begin{align*}f(x) = x^3\end{align*} is one to one.
3) Is \begin{align*}g(x) = x  2\end{align*} one to one?
Answers
1) Algebraic Test for 11 functions: if f(a) = f(b) implies that a = b, then f is 11
 \begin{align*}\therefore\end{align*} if \begin{align*}g(x) = 3x  2\end{align*} is 11, then \begin{align*}g(a) = g(b) \to a = b\end{align*}
 Test: \begin{align*}g(a) = g(b)\end{align*}
 \begin{align*}3a  2 = 3b  2\end{align*}
 \begin{align*}3a = 3b\end{align*}
 \begin{align*}a = b\end{align*}
 \begin{align*}\therefore 3x  2\end{align*} is \begin{align*}11\end{align*}
2) Graph the equation:
 This is the parent function of the cubic function family. Each x value has one unique yvalue that is not used by any other xelement. Since that is the definition of a 1:1 function, this function is 1:1.
3) Graph the equation:
 This absolute value function has yvalues that are paired with more than one xvalue, such as (4, 2) and (0, 2). This function is not 11. Note that this function also fails the horizontal line test used in Q 2.
Practice
 Describe the onetoone Horizontal Line Test
 Describe the onetoone Algebraic Test
Which functions are onetoone?
 \begin{align*}{ (3, 28), (4, 29), (4, 30), (6, 31) }\end{align*}
 \begin{align*}{ (4, 5), (9, 6), (7, 8), (23, 5) }\end{align*}
 \begin{align*}{ (8, 18), (33, 4), (5, 16), (7, 19) }\end{align*}
For the following to be a one to one function, X cannot be what values?
 \begin{align*}{ (9, 12), (35, 6), (7, 18), (12, X) }\end{align*}
 \begin{align*}{ (20, 21) (21, 14), (110, 112), (X, 7) }\end{align*}
Are the following onetoone functions?
 \begin{align*}f(x) = x^2\end{align*}
 \begin{align*}f(x) = x^3\end{align*}
 \begin{align*}f(x) = \frac{1}{x}\end{align*}
 \begin{align*}f(x) = x^n  x, n>0\end{align*}
 \begin{align*}x = y^2 + 2\end{align*}
Determine if the relations below are functions, oneto one functions or neither:
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Term  Definition 

11 function  A function is 11 if its inverse is also a function. 
Horizontal Line Test  The horizontal line test says that if a horizontal line drawn anywhere through the graph of a function intersects the function in more than one location, then the function is not onetoone and not invertible. 
inverse  Inverse functions are functions that 'undo' each other. Formally: and are inverse functions if . 
inverse function  Inverse functions are functions that 'undo' each other. Formally and are inverse functions if . 
invertible  A function is invertible if it has an inverse. 
Onetoone  A function is onetoone if its inverse is also a function. 
Vertical Line Test  The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function. 
Image Attributions
Here you will learn about 1 to 1 functions and how to identify their inverses.