## FG1 Functions and relations

A relation is a set of ordered pairs.

### Relations

A relation is a set of ordered pairs. For example $$(1,2),(2,6),(3,4),(x,y)$$ are ordered pairs and $$\left\{ (1,2),(2,6),(3,4),(x,y)\right\}$$is a relation.

The domain of a relation is the set of first elements or the $$x$$-values of the ordered pairs.

For the above ordered pairs the domain, dom = $$\left\{ 1,2,3,x\right\}$$.

The range of a relation is the set of second elements or the $$y$$-values of the ordered pairs. For the above ordered pairs the range, ran = $$\left\{ 2,4,6,y\right\}$$.

There is often a rule that links the domain and range.

For example: 1 The symbol $$\in$$ means “is in” or “is an element of” , the symbol $$\mathbb{R}$$ stands for the set of real numbers. The expression \begin{align*} x\in & \mathbb{R} \end{align*} means that “x is an element of the set of real numbers”. That is, $$x$$ is a real number. $S=\left\{ \left(x,y\right):y>x,\,x\in\mathbb{R}\right\}$

This relation, called S, consists of the set of all ordered pairs ($$x$$ and $$y$$), where the $$y$$ value is greater than the $$x$$ value and where $$x$$ must be a real number.

Note that a relation is defined by its rule (in this case $$y>x$$) and its domain (in this case $$x\in\mathbb{R}$$).2 If the domain is not given then we assume the largest possible domain.

#### Example 1

Sketch the graph of the following relation and state the domain and range:

$\left\{ \left(x,y\right):y=x^{2}\right\} .$

In this example the rule joining the set of ordered pairs ($$x,y$$) is $$y=x^{2}$$.

$$x$$ can be any real number. Domain is $$\mathbb{R}.$$

$$y$$ must be greater than or equal to zero. Range is $$\left\{ y:y\geq0\right\}$$

#### Example 2

Sketch the graph of $$x^{2}+y^{2}=4$$. State the Domain and Range of this relation.

In this example the rule joining the set of ordered pairs ($$x,y$$) is $$x^{2}+y^{2}=4$$.

From the graph it can be seen that the domain is $$\left\{ x:-2\leq x\leq2\right\}$$ and the range is $$\left\{ y:-2\leq y\leq2\right\} .$$

#### Example 3

Sketch the graph of $$\left\{ (x,y):2x+3y=6,x\geq0\right\}$$ and state the domain and range of this relation.

In this example the rule joining the set of ordered pairs $$(x,y)$$ is $$2x+3y=6$$.

The restriction $$x\geq0$$ is placed on the domain.

The domain is $$\left\{ x:x\geq0\right\}$$ as is specified in the statement of the relation.

The range is $$\left\{ y:y\leq2\right\}$$ as can be seen from the graph.

The rule of a relation may be thought of as: $$\textrm{DOMAIN\rightarrow\textrm{RULE\rightarrow\textrm{RANGE.} } }$$ Values taken from the domain produce values for the range, after passing through the rule that defines the relation.3 See Exercise 1

### Functions

From some of the previous examples it can be seen that some values in the domain ($$x$$ values) may have many, even an infinite number of corresponding values in the range ($$y$$ values).

A function is a special type of relation. Each point in the domain of a function has a unique value in the range. Every value of $$x$$ may have only one value of $$y$$ .

#### Examples

1. The relation $$\left\{ (-1,2),(-1,4),(1,6),(2,8),(3,10)\right\}$$ is not a function because the value $$x=-1$$ has two corresponding $$y$$ values $$(2\textrm{ and}$$ $$4)$$.

2. The relation $$\left\{ (-1,1),(0,2),(1,3),(2,5),(3,7)\right\}$$ is a function because for each $$x$$ value there is only one corresponding $$y$$ value.

3. $$F=\left\{ \left(x,y\right):y=\sin x,x\in R\right\}$$

If we choose any possible value of $$x,$$ there exists only one corresponding value of $$y$$. Therefore, the relation $$F$$ is a function.

Another way of writing this function is with mapping notation.

$f:X\rightarrow Y,\textrm{where f(x)=\sin x. }$

(The domain, $$X$$, is mapped onto the range, $$Y$$, using the rule $$f(x)=\sin x$$ )

If only the rule is given then we assume that the domain is $$\mathbb{R}$$.

#### Vertical Line Test

When relations are represented graphically, a vertical line test may be applied to decide if they are functions.

If a vertical line crosses the graph more than once, then it is not a function, as an $$x$$ value has more than one $$y$$ value.

The graph on the left is not a function (the vertical line crosses the graph more than once), the graph on the right is a function (vertical line only crosses the graph once). 4 See Exercise 2

### Implied Domain

If only the rule of the function is given, then we assume that the domain is $$\mathbb{R}$$ (the set of real numbers) unless otherwise defined implicitly by the function.

#### Examples

1. If a function involves a square root, the domain, in the real number system, is restricted to those values of $$x$$ that result in a non-negative number under the square root sign.
So, the domain of the function $$y=+\sqrt{x-4}$$ is restricted such that $$x-4\geq0$$ ; the domain is $$\left\{ x:x\geq4\right\}$$.

2. The domain of the function $$y=+\sqrt{9-x^{2}}$$ is restricted such that $$9-x^{2}\geq0$$ ; the domain is $$\left\{ x:-3\leq x\leq3\right\}$$.

3. If the function involves a fraction, the value in the denominator must not equal zero.
So, the domain of the function $$y=\dfrac{3}{x+5}$$ is restricted such that $$x+5\neq0$$ ; the domain is $$\left\{ x:x\neq-5\right\}$$.5 The domain may be written as \begin{align*} \left\{ x:x\in\mathbb{R}\setminus\left\{ -5\right\} \right\} . \end{align*} Here $$\mathbb{R}\setminus\left\{ -5\right\}$$ is the set of real numbers excluding $$-5$$.

4. The domain of the function $$y=\dfrac{3}{2x-8}$$ is restricted such that 2$$x-8\neq0$$ ; the domain is $$\left\{ x:x\neq4\right\}$$ or $$\left\{ x:x\in\mathbb{R}\setminus\left\{ 4\right\} \right\}$$. 6 See Exercise 3

### Exercises

#### Exercise 1.

State the domain and range of the following relations

1. $$\left\{ \left(-2,1\right),\left(0,2\right),\left(2,5\right),\left(2,7\right),\left(3,9\right)\right\}$$

2. $$\left\{ \left(4,1\right),\left(5,2\right),\left(6,3\right)\right\}$$

3. $$\left\{ \left(x,y\right):x^{2}+y^{2}=25\right\}$$

4. $$\left\{ \left(x,y\right):2y=6-5x,x\geq2\right\}$$

#### Exercise 2.

Which of the following relations are functions?

1. $$\left\{ \left(x,y\right):y=2x+4\right\}$$

2. $$\left\{ \left(x,y\right):y=4-x^{2}\right\}$$

3. $$\left\{ \left(x,y\right):x^{2}+y^{2}=36\right\}$$

4. $$\left\{ \left(x,y\right):y=7\right\}$$

5. $$\left\{ \left(x,y\right):x=-2\right\}$$

6. $$\left\{ \left(x,y\right):y=-\sqrt{4-x^{2}}\right\}$$

#### Exercise 3.

State the domain of the following functions.

1. $$\left\{ \left(x,y\right):y=x+2\right\}$$

2. $$\left\{ \left(x,y\right):y=4-x^{2}\right\}$$

3. $$\left\{ \left(x,y\right):y=+\sqrt{4-x}\right\}$$

4. $$\left\{ \left(x,y\right):y=\dfrac{3}{x+2}\right\}$$

5. $$\left\{ \left(x,y\right):y=\dfrac{5}{\sqrt{x-7}}\right\}$$

6. $$\left\{ \left(x,y\right):y=\dfrac{1}{x+2}-\dfrac{3}{x-4}\right\}$$

#### Exercise 1.

1. domain = $$\left\{ -2,0,2,3\right\}$$ range = $$\left\{ 1,2,5,7,9\right\}$$

2. domain = $$\left\{ 4,5,6\right\}$$ range = $$\left\{ 1,2,3\right\}$$

3. domain = $$\left\{ x:-5\leq x\leq5\right\}$$ range = $$\left\{ y:-5\leq y\leq5\right\}$$

4. domain = $$\left\{ x:x\geq2\right\}$$ range = $$\left\{ y:y\leq-2\right\}$$

#### Exercise 2.

(a), (b), (d), (f)

#### Exercise 3.

State the domain of the following functions.

1. $$\mathbb{R}$$

2. $$\mathbb{R}$$

3. $$\left\{ x:x\leq4\right\}$$

4. $$\left\{ x:x\neq-2\right\}$$

5. $$\left\{ x:x>7\right\}$$

6. $$\left\{ x:x\in\mathbb{R}\setminus\left\{ -4\right\} \right\}$$

What's next... FG2 Interval notation

Keywords: