
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 $$\PageIndex{1}$$.

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.