6.3: One-to-One Functions

We distinguish two special families of functions: the one-to-one functions and the onto functions. We shall discuss one-to-one functions in this section, and onto functions in the next.

Definition: Injection

A function ${f}:{A}\to{B}$ is said to be one-to-one if

$$x_1\neq x_2 \Rightarrow f(x_1)\neq f(x_2) \nonumber$$

for all elements $x_1,x_2\in A$. A one-to-one function is also called an injection, and we call a function injective if it is one-to-one. A function that is not one-to-one is referred to as many-to-one.

Any well-defined function is either one-to-one or many-to-one. A function cannot be one-to-many because no element can have multiple images. The difference between one-to-one and many-to-one functions is whether there exist distinct elements that share the same image. There are no repeated images in a one-to-one function.

Example $\PageIndex{1}\label{eg:oneonefcn-01}$

The identity function on any nonempty set $A$ $${i_A}:{A}\to{A}, \qquad i_A(x)=x, \nonumber$$ maps any element back to itself. It is clear that all identity functions are one-to-one.

Example $\PageIndex{2}\label{eg:oneonefcn-02}$

The function $h : {A}\to{A}$ defined by $h(x)=c$ for some fixed element $c\in A$, is an example of a constant function. It is a function with only one image. This is the exact opposite of an identity function. It is clearly not one-to-one unless $|A|=1$.

For domains with a small number of elements, one can use inspection on the images to determine if the function is one-to-one. This becomes impossible if the domain contains a larger number of elements.

In practice, it is easier to use the contrapositive of the definition to test whether a function is one-to-one:

$$f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \nonumber$$

Example $\PageIndex{3}\label{eg:oneonefcn-03}$

Is the function $f : {\mathbb{R}}\to{\mathbb{R}}$ defined by $f(x)=3x+2$ one-to-one?

Solution

Assume $f(x_1)=f(x_2)$, which means $$3x_1+2 = 3x_2+2. \nonumber$$ Thus $3x_1=3x_2$, which implies that $x_1=x_2$. Therefore $f$ is one-to-one.

exercise $\PageIndex{1}\label{he:oneonefcn-01}$

Determine whether the function $g : {\mathbb{R}}\to{\mathbb{R}}$ defined by $g(x)=5-7x$ is one-to-one.

exercise $\PageIndex{2}\label{he:oneonefcn-02}$

Determine whether the function $h : {[2,\infty)}\to{\mathbb{R}}$ defined by $h(x)=\sqrt{x-2}$ is one-to-one.

Interestingly, sometimes we can use calculus to determine if a real function is one-to-one. A real function $f$ is increasing if $$x_1 < x_2 \Rightarrow f(x_1) < f(x_2), \nonumber$$ and decreasing if $$x_1 < x_2 \Rightarrow f(x_1) > f(x_2). \nonumber$$ Obviously, both increasing and decreasing functions are one-to-one. From calculus, we know that

• A function is increasing over an open interval $(a,b)$ if $f'(x)>0$ for all $x\in(a,b)$.
• A function is decreasing over an open interval $(a,b)$ if $f'(x)<0$ for all $x\in(a,b)$.

Therefore, if the derivative of a function is always positive, or always negative, then the function must be one-to-one.

Example $\PageIndex{4}\label{eg:oneonefcn-04}$

The function $p : {\mathbb{R}}\to{\mathbb{R}}$ defined by $$p(x) = 2x^3-5 \nonumber$$ is one-to-one, because $p'(x)=6x^2>0$ for any $x\in\mathbb{R}^*$. Likewise, the function $q:{\big(-\frac{\pi}{2},\frac{\pi}{2}\big)}\to{\mathbb{R}}$ defined by $$q(x) = \tan x \nonumber$$ is also one-to-one, because $q'(x) = \sec^2x > 0$ for any $x\in \big(-\frac{\pi}{2},\frac{\pi}{2}\big)$.

exercise $\PageIndex{3}\label{he:oneonefcn-03}$

Use both methods to show that the function $k:{(0,\infty)}\to{\mathbb{R}}$ defined by $k(x) = \ln x$ is one-to-one.

Example $\PageIndex{5}\label{eg:oneonefcn-05}$

The function $h : {\mathbb{R}}\to{\mathbb{R}}$ given by $h(x)=x^2$ is not one-to-one because some of its images are identical. For example, $h(3) = h(-3) =9$. It is a many-to-one function. Likewise, the absolute value function $|x|$ is not one-to-one.

The functions $p:{[\,0,\infty)}\to{\mathbb{R}}$ defined by $p(x)=x^2$ and $q:{[\,0,\infty)}\to{\mathbb{R}}$ defined by $q(x)=|x|$ are one-to-one. Whether a function is one-to-one depends not only on its formula, but also on its domain. Consequently, sometimes we may be able to convert a many-to-one function into a one-to-one function by modifying its domain.

Example $\PageIndex{6}\label{eg:onetoone}$

Construct a one-to-one function from $[\,1,3\,]$ to $[\,2,5\,]$.

Solution

There are many possible solutions. In any event, start with a graph. We can use a straight line graph. The domain $[\,1,3\,]$ lies on the $x$-axis, and the codomain $[\,2,5\,]$ lies on the $y$-axis. Hence the graph should cover the boxed region in Figure $\PageIndex{1}$.

Figure $\PageIndex{1}$: Three candidates for one-to-one functions from [1,3] to [2,5].

All three graphs do not produce duplicate images. We need to cover all $x$-values from 1 to 3 in order for the function to be well-defined. This leaves only the first two graphs as legitimate examples.

To determine the formula for $f$, we need to derive the equation of the line. Take the first graph as our choice. The line joins the point $(1,2)$ to the point $(3,4)$. Thus, its equation is $$\frac{y-2}{x-1} = \frac{4-2}{3-1} = 1. \nonumber$$ The last step is to write the answer in the form of $f(x)=\ldots\,$. We have to express $y$ in terms of $x$. We find $y = x+1$. Hence, $$f : {[\,1,3\,]}\to{[\,2,5\,]}, \qquad f(x) = x+1 \nonumber$$ is an example of a one-to-one function.

exercise $\PageIndex{4}\label{he:oneonefcn-04}$

Construct a one-to-one function from $[\,1,3\,]$ to $[\,2,5\,]$ based on the second graph in Example 6.3.6.

exercise $\PageIndex{5}\label{he:oneonefcn-05}$

Construct a one-to-one function from $[\,3,8\,]$ to $[\,2,5\,]$.

example $\PageIndex{7}\label{eg:gmod43}$

Determine whether the function $g : {\mathbb{Z}_{43}}\to{\mathbb{Z}_{43}}$ defined by $$g(x) \equiv 11x-5 \pmod{43} \nonumber$$ is one-to-one.

Solution

Assume $g(x_1)=g(x_2)$. This means $$11x_1 - 5 \equiv 11x_2 - 5 \pmod{43}, \nonumber$$ which implies $$11x_1 \equiv 11x_2 \pmod{43}. \nonumber$$ Notice that $4\cdot11=44\equiv1$ (mod 43), hence $11^{-1}\equiv4$ (mod 43). Multiplying 4 to both sides of the last congruence yields $$44 x_1 \equiv 44 x_2 \pmod{43}, \nonumber$$ which is equivalent to, since $44\equiv1$ (mod 43), $$x_1 \equiv x_2 \pmod{43}. \nonumber$$ Therefore, $x_1=x_2$ in $\mathbb{Z}_{43}$. This proves that $g$ is one-to-one.

exercise $\PageIndex{6}\label{he:oneonefcn-06}$

Is the function $h : {\mathbb{Z}_{15}}\to{\mathbb{Z}_{15}}$ defined by $$h(x) \equiv 4x-11 \pmod{15} \nonumber$$ a one-to-one function?

exercise $\PageIndex{7}\label{he:oneonefcn-07}$

Show that the function $k:{\mathbb{Z}_{15}}\to{\mathbb{Z}_{15}}$ defined by $$k(x) \equiv 5x-11 \pmod{15} \nonumber$$ is not one-to-one by finding $x_1\neq x_2$ such that $k(x_1)=k(x_2)$.

Example $\PageIndex{8}\label{eg:oneonefcn-08}$

In the last exercise, we should not rely on the non-existence of $5^{-1}$ in $\mathbb{Z}_{15}$ to prove that $k$ is not one-to-one. One must consider the interaction between the domain, the codomain, and the definition of the function. For example, despite the fact that $5^{-1}$ does not exist in $\mathbb{Z}_{15}$, the function $p :{\mathbb{Z}_3}\to{\mathbb{Z}_{15}}$ defined by $$p(x) \equiv 5x-11 \pmod{15} \nonumber$$ is one-to-one, because $p(0)=4$, $p(1)=9$, and $p(2)=14$ are distinct images.

The last example illustrates the trickiness in a function with different moduli in its domain and codomain. Use caution when you deal with such functions! Sometimes, infinite sets also pose a challenge. Because there is an infinite supply of elements, we may obtain results that appear to be impossible for finite sets.

Example $\PageIndex{9}\label{eg:oneonefcn-09}$

The function $f : {\mathbb{Z}}\to{\mathbb{Z}}$ defined by $$f(n) = \cases{ \frac{n}{2} & if $n$ is even \cr \frac{n+1}{2} & if $n$ is odd \cr} \nonumber$$ is not one-to-one, because, for example, $f(0)=f(-1)=0$. The function $g : {\mathbb{Z}}\to{\mathbb{Z}}$ defined by $$g(n) = 2n \nonumber$$ is one-to-one, because if $g(n_1)=g(n_2)$, then $2n_1=2n_2$ implies that $n_1=n_2$.

exercise $\PageIndex{8}\label{he:oneonefcn-08}$

Show that the function $h : {\mathbb{Z}}\to{\mathbb{N}}$ defined by $$h(n) = \cases{ 2n+1 & if $n\geq0$, \cr -2n & if $n < 0$, \cr} \nonumber$$ is one-to-one.

Example $\PageIndex{10}\label{eg:oneonefcn-10}$

Let $A$ be the set of all married individuals from a monogamous community who are neither divorced nor widowed. Then the function $s:{A}\to{A}$ defined by $$s(x) = \mbox{ spouse of } x \nonumber$$ is one-to-one. The reason is, it is impossible to have $x_1\neq x_2$ and yet $s(x_1)=s(x_2)$.

Summary and Review

• A function $f$ is said to be one-to-one if $f(x_1) = f(x_2) \Rightarrow x_1=x_2$.
• No two images of a one-to-one function are the same.
• To show that a function $f$ is not one-to-one, all we need is to find two different $x$-values that produce the same image; that is, find $x_1\neq x_2$ such that $f(x_1)=f(x_2)$.

Exercise $\PageIndex{1}\label{ex:oneonefcn-01}$

Which of the following functions are one-to-one? Explain.

• $f : {\mathbb{R}}\to{\mathbb{R}}$, $f(x)=x^3-2x^2+1$.
• $g : {[\,2,\infty)}\to{\mathbb{R}}$, $f(x)=x^3-2x^2+1$.

Exercise $\PageIndex{2}\label{ex:oneonefcn-02}$

Which of the following functions are one-to-one? Explain.

• $p :{\mathbb{R}}\to{\mathbb{R}}$, $h(x)=e^{1-2x}$.
• $q :{\mathbb{R}}\to{\mathbb{R}}$, $p(x)=|1-3x|$.

Exercise $\PageIndex{3}\label{ex:oneonefcn-03}$

Construct a one-to-one function $f : {(1,3)}\to{(2,5)}$ so that $f : {[\,1,3)}\to{[\,2,5)}$ is still one-to-one.

Exercise $\PageIndex{4}\label{ex:oneonefcn-04}$

Construct a one-to-one function $g : {[\,2,5)}\to{(1,4\,]}$.

Exercise $\PageIndex{5}\label{ex:oneonefcn-05}$

Determine which of the following are one-to-one functions.

1. $f : {\mathbb{Z}}\to{\mathbb{Z}}$; $f(n)=n^3+1$
2. $g : {\mathbb{Q}}\to{\mathbb{Q}}$; $g(x)=n^2$
3. $h : {\mathbb{R}}\to{\mathbb{R}}$; $h(x)=x^3-x$
4. ${k} : {\mathbb{R}}\to{\mathbb{R}}$; $k(x)=5^x$

Exercise $\PageIndex{6}\label{ex:oneonefcn-06}$

Determine which of the following are one-to-one functions.

1. $p :{\wp(\{1,2,3,\ldots,n\})}\to{\{0,1,2,\ldots,n\}}$; $p(S)=|S|$
2. $q :{\wp(\{1,2,3,\ldots,n\})}\to{\wp(\{1,2,3,\ldots,n\})}$; $q(S)=\overline{S}$

Exercise $\PageIndex{7}\label{ex:oneonefcn-07}$

Determine which of the following functions are one-to-one.

1. ${f_1}:{\{1,2,3,4,5\}}\to{\{a,b,c,d\}}$; $f_1(1)=b$, $f_1(2)=c$, $f_1(3)=a$, $f_1(4)=a$, $f_1(5)=c$
2. ${f_2}:{\{1,2,3,4\}}\to{\{a,b,c,d,e\}}$; $f_2(1)=c$, $f_2(2)=b$, $f_2(3)=a$, $f_2(4)=d$
3. ${f_3}:{\mathbb{Z}}\to{\mathbb{Z}}$; $f_5(n)=-n$
4. $f_4: \mathbb{Z} \rightarrow \mathbb{Z}$; $f_4(n) = \cases{ 2n & if $n < 0$, \cr -3n & if $n\geq0$,\cr}$

Exercise $\PageIndex{8}\label{ex:oneonefcn-08}$

Determine which of the following functions are one-to-one.

1. ${g_1}:{\{1,2,3,4,5\}}\to{\{a,b,c,d,e\}}$; $g_1(1)=b$, $g_1(2)=b$, $g_1(3)=b$, $g_1(4)=a$, $g_1(5)=d$
2. ${g_2}:{\{1,2,3,4,5\}}\to{\{a,b,c,d,e\}}$; $g_2(1)=d$, $g_2(2)=b$, $g_2(3)=e$, $g_2(4)=a$, $g_2(5)=c$
3. $g_3: \mathbb{N} \rightarrow \mathbb{N}$; $g_3 (n) = \cases{ \frac{n+1}{2} & if $n$ is odd \cr \frac{n}{2} & if $n$ is even \cr}$
4. $g_4: \mathbb{N} \rightarrow \mathbb{N}$; $g_4 (n) = \cases{ n+1 & if $n$ is odd \cr n-1 & if $n$ is even \cr}$

Exercise $\PageIndex{9}\label{ex:oneonefcn-09}$

List all the one-to-one functions from $\{1,2\}$ to $\{a,b,c,d\}$.

Hint

List the images of each function.

Exercise $\PageIndex{10}\label{ex:oneonefcn-10}$

Is it possible to find a one-to-one function from $\{1,2,3,4\}$ to $\{1,2\}$? Explain.

Exercise $\PageIndex{11}\label{ex:oneonefcn-11}$

Determine which of the following functions are one-to-one.

1. $f : {\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}$; $h(n)\equiv 3n$ (mod 10).
2. $g : {\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}$; $g(n)\equiv 5n$ (mod 10).
3. $h : {\mathbb{Z}_{36}}\to{\mathbb{Z}_{36}}$; $h(n)\equiv 3n$ (mod 36).

Exercise $\PageIndex{12}\label{ex:oneonefcn-12}$

Determine which of the following functions are one-to-one.

1. $r:{\mathbb{Z}_{36}}\to{\mathbb{Z}_{36}}$; $r(n)\equiv 5n$ (mod 36).
2. $s:{\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}$; $s(n)\equiv n+5$ (mod 10).
3. $t:{\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}$; $t(n)\equiv 3n+5$ (mod 10).

Exercise $\PageIndex{13}\label{ex:oneonefcn-13}$

Determine which of the following functions are one-to-one.

1. $\alpha:{\mathbb{Z}_{12}}\to{\mathbb{Z}_{ 7}}$; $\alpha(n)\equiv 2n$ (mod 7).
2. $\beta :{\mathbb{Z}_{ 8}}\to{\mathbb{Z}_{12}}$; $\beta (n)\equiv 3n$ (mod 12).
3. $\gamma:{\mathbb{Z}_{ 6}}\to{\mathbb{Z}_{12}}$; $\gamma(n)\equiv 2n$ (mod 12).
4. $\delta:{\mathbb{Z}_{12}}\to{\mathbb{Z}_{36}}$; $\delta(n)\equiv 6n$ (mod 36).

Exercise $\PageIndex{14}\label{ex:oneonefcn-14}$

Give an example of a one-to-one function $f$ from $\mathbb{N}$ to $\mathbb{N}$ that is not the identity function.