Learning Lab

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.33 See Exercise 1

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). 44 See Exercise 2

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 \(\left\{ x:x\neq-5\right\}\).55 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 \(\left\{ x:x\in\mathbb{R}\setminus\left\{ 4\right\} \right\}\). 66 See Exercise 3

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. \(\left\{ x:x\in\mathbb{R}\setminus\left\{ -4\right\} \right\}\)