Relations and Functions

This chapter covers ordered pairs, Cartesian product, relations, domain, range, codomain, functions, real-valued functions, and basic function graphs.

Chapter overview

Relations and functions describe how elements of one set are paired with elements of another set. This chapter forms the base for calculus, coordinate geometry, inverse trigonometry, and many areas of higher mathematics.

1. Ordered pair

An ordered pair is written as:

$$(a,b)$$

Here order matters.

$$(a,b)\ne(b,a) \quad\text{in general}$$

Equality of ordered pairs:

$$(a,b)=(c,d) \iff a=c,\ b=d$$

2. Cartesian product

If $A$ and $B$ are non-empty sets:

$$A\times B=\{(a,b):a\in A,\ b\in B\}$$

Example:

$$A=\{1,2\},\quad B=\{x,y\}$$

Then:

$$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$

Number of ordered pairs:

$$n(A\times B)=n(A)\cdot n(B)$$

Usually:

$$A\times B\ne B\times A$$

3. Relation

A relation $R$ from $A$ to $B$ is a subset of $A\times B$.

$$R\subseteq A\times B$$

If:

$$(a,b)\in R$$

then $a$ is related to $b$.

4. Domain and range of relation

Domain:

$$\text{Domain}(R)=\{a\in A:(a,b)\in R\text{ for some }b\in B\}$$

Range:

$$\text{Range}(R)=\{b\in B:(a,b)\in R\text{ for some }a\in A\}$$

Example:

$$R=\{(1,2),(2,4),(3,6)\}$$

Then:

$$\text{Domain}(R)=\{1,2,3\}$$ $$\text{Range}(R)=\{2,4,6\}$$

5. Function

A function $f$ from $A$ to $B$ is a relation in which every element of $A$ has exactly one image in $B$.

$$f:A\rightarrow B$$

If $x\in A$, then image of $x$ is:

$$f(x)$$

Function condition:

6. Domain, codomain, and range of function

For:

$$f:A\rightarrow B$$

$A$ is the domain and $B$ is the codomain.

Range:

$$\text{Range}(f)=\{f(x):x\in A\}$$

Important:

$$\text{Range}(f)\subseteq\text{Codomain}$$

7. Real-valued functions

A real-valued function has real numbers as outputs.

$$f:A\rightarrow R$$

Examples:

$$f(x)=x^2$$ $$g(x)=\frac{1}{x}$$ $$h(x)=\sqrt{x}$$

8. Domain restrictions

For rational functions, denominator cannot be zero.

If:

$$f(x)=\frac{1}{x-2}$$

then:

$$x\ne2$$

So:

$$\text{Domain}=R-\{2\}$$

For square root:

$$f(x)=\sqrt{x-3}$$

Need:

$$x-3\ge0$$

Therefore:

$$x\ge3$$

9. Types of simple functions

Identity function

$$f:R\rightarrow R,\quad f(x)=x$$

Constant function

$$f:R\rightarrow R,\quad f(x)=c$$

Polynomial function

$$f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$$

Rational function

$$f(x)=\frac{p(x)}{q(x)},\quad q(x)\ne0$$

Modulus function

$$|x|= \begin{cases} x, & x\ge0\\ -x, & x<0 \end{cases}$$

Signum function

$$\operatorname{sgn}(x)= \begin{cases} 1, & x>0\\ 0, & x=0\\ -1, & x<0 \end{cases}$$

Greatest integer function

$$f(x)=[x]$$

where $[x]$ is the greatest integer less than or equal to $x$.

10. Algebra of real functions

If $f$ and $g$ are real functions, then:

$$(f+g)(x)=f(x)+g(x)$$ $$(f-g)(x)=f(x)-g(x)$$ $$(fg)(x)=f(x)g(x)$$ $$\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)},\quad g(x)\ne0$$

11. Function mapping diagram

This represents:

$$f(x)=2x$$

for:

$$A=\{1,2,3\}$$

12. Solved examples

Example 1

If:

$$A=\{1,2\},\quad B=\{3,4\}$$

then:

$$A\times B=\{(1,3),(1,4),(2,3),(2,4)\}$$

Example 2

Find domain of:

$$f(x)=\frac{1}{x^2-9}$$

Need:

$$x^2-9\ne0$$ $$(x-3)(x+3)\ne0$$

So:

$$x\ne3,\quad x\ne-3$$

Domain:

$$R-\{-3,3\}$$

Example 3

If:

$$f(x)=x^2,\quad g(x)=x+1$$

then:

$$(f+g)(x)=x^2+x+1$$ $$(fg)(x)=x^2(x+1)=x^3+x^2$$

13. Common mistakes

• Thinking every relation is a function.
• Forgetting every input must have exactly one output.
• Confusing codomain and range.
• Forgetting denominator cannot be zero.
• Forgetting square root expression must be non-negative.
• Treating $A\times B$ and $B\times A$ as always equal.

14. Quick revision