7.1: One-to-one functions
- Page ID
- 48989
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We have seen that some functions \(f\) may have the same outputs for different inputs. For example for \(f(x)=x^2\), the inputs \(x=2\) and \(x=-2\) have the same output \(f(2)=4\) and \(f(-2)=4\). A function is one-to-one, precisely when this is not the case.
A function \(f\) is called one-to-one (or injective), if two different inputs \(x_1\neq x_2\) always have different outputs \(f(x_1)\neq f(x_2)\).
As was noted above, the function \(f(x)=x^2\) is not one-to-one, because, for example, for inputs \(2\) and \(-2\), we have the same output
\[f(-2)=(-2)^2=4, \quad\quad f(2)=2^2=4 \nonumber \]
On the other hand, \(g(x)=x^3\) is one-to-one, since, for example, for inputs \(-2\) and \(2\), we have different outputs:
\[g(-2)=(-2)^3=-8, \quad\quad g(2)=2^3=8 \nonumber \]
The difference between the functions \(f\) and \(g\) can be seen from their graphs.
The graph of \(f(x)=x^2\) on the left has for different inputs (\(x_0\) and \(-x_0\)) the same output (\(y_0=(x_0)^2=(-x_0)^2\)). This is shown in the graph since the horizontal line at \(y_0\) intersects the graph at two different points. In general, two inputs that have the same output \(y_0\) give two points on the graph which also lie on the horizontal line at \(y_0\).
Now, the graph of \(g(x)=x^3\) on the right intersects with a horizontal line at some \(y_0\) only once. This shows that for two different inputs, we can never have the same output \(y_0\), so that the function \(g\) is one-to-one.
We can summarize the observation of the last example in the following statement.
A function is one-to-one exactly when every horizontal line intersects the graph of the function at most once.
Which of the following are or represent one-to-one functions?
- \(f(x)=-x^3+6x^2-13x+12\)
- \(f(x)=x^3-2x^2+3\)
Solution
We use the horizontal line test to see which functions are one-to-one. For (a) and (b), we see that the functions are not one-to-one since there is a horizontal line that intersects with the graph more than once:
For (c), using the calculator to graph the function \(f(x)=-x^3+6x^2-13x+12\), we see that all horizontal lines intersect the graph exactly once. Therefore the function in part (c) is one-to-one. The function in part (d) however has a graph that intersects some horizontal line in several points. Therefore \(f(x)=x^3-2x^2+3\) is not one-to-one:
