3.1.1: One-to-One Functions and Their Inverses
- Page ID
- 14368
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)One-to-One 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 x-axis, and different foods the restaurant specializes in on the y-axis. Any time a pizza restaurant was input into the function, it would output "pizza" as the specialized food.
Is this pizza restaurant function a one-to-one function? How can we tell?
One-to-One Functions and Their Inverses
Consider the function \(\ f(x)=x^{3}\), and its inverse \(\ f^{-1}(x)=\sqrt[3]{x}\).
The graphs of these functions are shown below:
The function f(x) = x3 is an example of a one-to-one function, which is defined as follows:
A function is one-to-one if and only if every element of its range corresponds to at most one element of its domain.
The function y = x2, however, is not one-to-one. 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 = x2, 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 = x2, the result is a graph that is a reflection over the line y = x, effectively rotating the original 90 degrees. Since x and y have swapped, the new function fails the vertical line test.
The function y = x2 is therefore not a one-to-one function. A function that is one-to-one 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.
Examples
Earlier, you were given a question about a pizza function.
Solution
"Pizza restaurants sell pizza" is a function. However, it is NOT a one-to-one function.
In order to be one-to-one, 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.
Graph the function \(\ f(x)=\frac{1}{3} x+2\). 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 \(\ f(x)=\frac{1}{3} x+2\).
In sum, a one-to-one function is invertible. That is, if we invert a one-to-one 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.
State the domain and range of the following function and its inverse: (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) = x3, 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 = x2 is not one-to-one, 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) = x2, with domain limited to real numbers ≥ 0. Then the inverse of the function is the square root function: \(\ f^{-1}(x)=\sqrt{x}\).
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.
Is g(x)=3x−2 a one-to-one function?
Solution
Algebraic test for one-to-one functions: if f(a) = f(b) implies that a = b, then f is one-to-one.
∴ if g(x)=3x−2 is one-to-one, then g(a)=g(b)→a=b
Test: g(a)=g(b)
3a−2=3b−2
3a=3b
a=b
∴3x−2 is one-to-one.
Use the horizontal line test to see if f(x)=x3 is one-to-one.
Solution
Graph the equation:
This is the parent function of the cubic function family. Each x value has one unique y-value that is not used by any other x-element. Since that is the definition of a one-to-one function, this function is one-to-one.
Is g(x)=|x−2| one-to-one?
Solution
Graph the equation:
This absolute value function has y-values that are paired with more than one x-value, such as (4, 2) and (0, 2). This function is not one-to-one. Note that this function also fails the horizontal line test used in Example 6.
Review
- Describe the one-to-one horizontal line test.
- Describe the one-to-one algebraic test.
Which functions are one-to-one?
- (3,28),(4,29),(4,30),(6,31)
- (4,5),(9,6),(7,8),(23,5)
- (8,18),(33,4),(5,16),(7,19)
For the following to be a one-to-one function, x cannot be what values?
- (9,12),(35,6),(7,18),(12,X)
- (20,21)(21,14),(110,112),(X,7)
Are the following one-to-one functions?
- f(x)=x2
- f(x)=x3
- f(x)=\(\ \frac{1}{x}\)
- f(x)=xn−x,n>0
- x=y2+2
Determine if the relations below are functions, one-to one functions or neither:
Vocabulary
| Term | Definition |
|---|---|
| 1-1 function | A function is 1-1 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 one-to-one and not invertible. |
| inverse | Inverse functions are functions that 'undo' each other. Formally: f(x) and g(x) are inverse functions if f(g(x))=g(f(x))=x. |
| inverse function | Inverse functions are functions that 'undo' each other. Formally f(x) and g(x) are inverse functions if f(g(x))=g(f(x))=x. |
| invertible | A function is invertible if it has an inverse. |
| One-to-one | A function is one-to-one 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. |