1.2: Relations and Functions (2024)

Definition: Relation

A relation is any set of ordered pairs, \((x,y)\). All the x-values in the ordered pairs together make up the domain. All the y-values in the ordered pairs together make up the range.

Example \(\PageIndex{1}\)

For the relation \({(1,1),(2,4),(3,9),(4,16),(5,25)}\):

1. Find the domain of the relation.
2. Find the range of the relation.

Answer

\[\begin{array} {ll} {} &{ {\{(1,1), (2,4), (3,9), (4,16), (5,25) }\} } \\ {ⓐ\text{ The domain is the set of all x-values of the relation.}} &{ {\{1,2,3,4,5}\} } \\ {ⓑ\text{ The range is the set of all y-values of the relation.}} &{ {\{1,4,9,16,25}\} } \\ \nonumber \end{array}\]

Example \(\PageIndex{2}\)

For the relation \({\{(1,1),(2,8),(3,27),(4,64),(5,125)}\}\):

1. Find the domain of the relation.
2. Find the range of the relation.

Answer a

\({\{1,2,3,4,5}\}\)

Answer b

\({\{1,8,27,64,125}\}\)

Example \(\PageIndex{3}\)

For the relation \({\{(1,3),(2,6),(3,9),(4,12),(5,15)}\}\):

1. Find the domain of the relation.
2. Find the range of the relation.

Answer a

\({\{1,2,3,4,5}\}\)

Answer b

\({\{3,6,9,12,15}\}\)

MAPPING

A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.

Example \(\PageIndex{4}\)

Use the mapping of the relation shown to

1. list the ordered pairs of the relation,
2. find the domain of the relation, and
3. find the range of the relation.

Answer

ⓐ The arrow shows the matching of the person to their birthday. We create ordered pairs with the person’s name as the x-value and their birthday as the y-value.

{(Alison, April 25), (Penelope, May 23), (June, August 2), (Gregory, September 15), (Geoffrey, January 12), (Lauren, May 10), (Stephen, July 24), (Alice, February 3), (Liz, August 2), (Danny, July 24)}

ⓑ The domain is the set of all x-values of the relation.

{Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, Danny}

ⓒ The range is the set of all y-values of the relation.

{January 12, February 3, April 25, May 10, May 23, July 24, August 2, September 15}

Example \(\PageIndex{5}\)

Use the mapping of the relation shown to

1. list the ordered pairs of the relation
2. find the domain of the relation
3. find the range of the relation.

Answer

ⓐ (Khanh Nguyen, kn68413), (Abigail Brown, ab56781), (Sumantha Mishal, sm32479), (Jose Hern and ez, jh47983)

ⓑ {Khanh Nguyen, Abigail Brown, Sumantha Mishal, Jose Hern and ez}

ⓒ {kn68413, ab56781, sm32479, jh47983}

Example \(\PageIndex{6}\)

Use the mapping of the relation shown to

1. list the ordered pairs of the relation
2. find the domain of the relation
3. find the range of the relation.

Answer

ⓐ (Maria, November 6), (Arm and o, January 18), (Cynthia, December 8), (Kelly, March 15), (Rachel, November 6)

ⓑ {Maria, Arm and o, Cynthia, Kelly, Rachel}

ⓒ{November 6, January 18, December 8, March 15}

Example \(\PageIndex{7}\)

Use the graph of the relation to

1. list the ordered pairs of the relation
2. find the domain of the relation
3. find the range of the relation.

Answer

ⓐ The ordered pairs of the relation are: \[{\{(1,5),(−3,−1),(4,−2),(0,3),(2,−2),(−3,4)}\}.\nonumber\]

ⓑ The domain is the set of all x-values of the relation: \(\quad {\{−3,0,1,2,4}\}\).

Notice that while \(−3\) repeats, it is only listed once.

ⓒ The range is the set of all y-values of the relation: \(\quad {\{−2,−1,3,4,5}\}\).

Notice that while \(−2\) repeats, it is only listed once.

Example \(\PageIndex{8}\)

Use the graph of the relation to

1. list the ordered pairs of the relation
2. find the domain of the relation
3. find the range of the relation.

Answer

a.\((−3,3),(−2,2),(−1,0),\), \((0,−1),(2,−2),(4,−4)\)

b.\({\{−3,−2,−1,0,2,4}\}\)

c.\({\{3,2,0,−1,−2,−4}\}\)

Example \(\PageIndex{9}\)

Use the graph of the relation to

1. list the ordered pairs of the relation
2. find the domain of the relation
3. find the range of the relation.

Answer

ⓐ \((−3,0),(−3,5),(−3,−6),\)
\((−1,−2),(1,2),(4,−4)\)
ⓑ \({\{−3,−1,1,4}\}\)
ⓒ \({\{−6,0,5,−2,2,−4}\}\)

Definition: Function

A function is a relation that assigns to each element in its domain exactly one element in the range.

Example \(\PageIndex{10}\)

Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.

1. \({\{(−3,27),(−2,8),(−1,1),(0,0),(1,1),(2,8),(3,27)}\}\)
2. \({\{(9,−3),(4,−2),(1,−1),(0,0),(1,1),(4,2),(9,3)}\}\)

Answer

ⓐ \({\{(−3,27),(−2,8),(−1,1),(0,0),(1,1),(2,8),(3,27)}\}\)

See Also

Checking if a table represents a function (video) | Khan Academy

(i) Each x-value is matched with only one y-value. So this relation is a function.

(ii) The domain is the set of all x-values in the relation.
The domain is: \({\{−3,−2,−1,0,1,2,3}\}\).

(iii) The range is the set of all y-values in the relation. Notice we do not list range values twice.
The range is: \({\{27,8,1,0}\}\).

ⓑ \({\{(9,−3),(4,−2),(1,−1),(0,0),(1,1),(4,2),(9,3)}\}\)

(i) The x-value 9 is matched with two y-values, both 3 and \(−3\). So this relation is not a function.

(ii) The domain is the set of all x-values in the relation. Notice we do not list domain values twice.
The domain is: \({\{0,1,2,4,9}\}\).

(iii) The range is the set of all y-values in the relation.
The range is: \({\{−3,−2,−1,0,1,2,3}\}\).

Example \(\PageIndex{11}\)

Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the function.

1. \({\{(−3,−6),(−2,−4),(−1,−2),(0,0),(1,2),(2,4),(3,6)}\}\)
2. \({\{(8,−4),(4,−2),(2,−1),(0,0),(2,1),(4,2),(8,4)}\}\)

Answer

ⓐ Yes; \({\{−3,−2,−1,0,1,2,3}\}\);
\({\{−6,−4,−2,0,2,4,6}\}\)
ⓑ No; \({\{0,2,4,8}\}\);
\({\{−4,−2,−1,0,1,2,4}\}\)

Example \(\PageIndex{12}\)

Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.

1. \({\{(27,−3),(8,−2),(1,−1),(0,0),(1,1),(8,2),(27,3)}\}\)
2. \({\{(7,−3),(−5,−4),(8,−0),(0,0),(−6,4),(−2,2),(−1,3)}\}\)

Answer

ⓐ No; \({\{0,1,8,27}\}\);
\({\{−3,−2,−1,0,2,2,3}\}\)
ⓑ Yes; \({\{7,−5,8,0,−6,−2,−1}\}\);
\({\{−3,−4,0,4,2,3}\}\)

Example \(\PageIndex{13}\)

Use the mapping to

1. determine whether the relation is a function
2. find the domain of the relation
3. find the range of the relation.

Answer

ⓐ Both Lydia and Marty have two phone numbers. So each x-value is not matched with only one y-value. So this relation is not a function.

ⓑ The domain is the set of all x-values in the relation. The domain is: {Lydia, Eugene, Janet, Rick, Marty}

ⓒ The range is the set of all y-values in the relation. The range is:

\({\{321-549-3327, 427-658-2314, 321-964-7324, 684-358-7961, 684-369-7231, 798-367-8541}\}\)

Example \(\PageIndex{14}\)

Use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation.

Answer

ⓐ no ⓑ {NBC, HGTV, HBO} ⓒ {Ellen Degeneres Show, Law and Order, Tonight Show, Property Brothers, House Hunters, Love it or List it, Game of Thrones, True Detective, Sesame Street}

Example \(\PageIndex{15}\)

Use the mapping to

1. determine whether the relation is a function
2. find the domain of the relation
3. find the range of the relation.

Answer

ⓐ No ⓑ {Neal, Krystal, Kelvin, George, Christa, Mike} ⓒ {123-567-4839 work, 231-378-5941 cell, 743-469-9731 cell, 567-534-2970 work, 684-369-7231 cell, 798-367-8541 cell, 639-847-6971 cell}

Example \(\PageIndex{16}\)

Determine whether each equation is a function.

1. \(2x+y=7\)
2. \(y=x^2+1\)
3. \(x+y^2=3\)

Answer

ⓐ \(2x+y=7\)

For each value of x, we multiply it by \(−2\) and then add 7 to get the y-value

For example, if \(x=3\):

We have that when \(x=3\), then \(y=1\). It would work similarly for any value of x. Since each value of x, corresponds to only one value of y the equation defines a function.

ⓑ \(y=x^2+1\)

For each value of x, we square it and then add 1 to get the y-value.

For example, if \(x=2\):

We have that when \(x=2\), then \(y=5\). It would work similarly for any value of x. Since each value of x, corresponds to only one value of y the equation defines a function.

ⓒ

Isolate the y term.
Let’s substitute \(x=2\).
This give us two values for y.	\(y=1\space y=−1\)

We have shown that when \(x=2\), then \(y=1\) and \(y=−1\). It would work similarly for any value of x. Since each value of x does not corresponds to only one value of y the equation does not define a function.

Example \(\PageIndex{17}\)

Determine whether each equation is a function.

1. \(4x+y=−3\)
2. \(x+y^2=1\)
3. \(y−x^2=2\)

Answer

ⓐ yes ⓑ no ⓒ yes

Example \(\PageIndex{18}\)

Determine whether each equation is a function.

1. \(x+y^2=4\)
2. \(y=x^2−7\)
3. \(y=5x−4\)

Answer

ⓐ no ⓑ yes ⓒ yes

Definition: Function Notation

For the function \(y=f(x)\)

\[\begin{array} {l} {f\text{ is the name of the function}} \\{x \text{ is the domain value}} \\ {f(x) \text{ is the range value } y \text{ corresponding to the value } x} \\ \nonumber \end{array}\]

We read \(f(x)\) as \(f\) of \(x\) or the value of \(f\) at \(x\).

INDEPENDENT AND DEPENDENT VARIABLES

For the function \(y=f(x)\),

\[\begin{array} {l} {x \text{ is the independent variable as it can be any value in the domain}} \\ {y \text{ the dependent variable as its value depends on } x} \\ \nonumber \end{array}\]

Example \(\PageIndex{19}\)

For the function \(f(x)=2x^2+3x−1\), evaluate the function.

1. \(f(3)\)
2. \(f(−2)\)
3. \(f(a)\)

Answer

ⓐ

To evaluate \(f(3)\), substitute 3 for x.
Simplify.

ⓑ

Simplify.

ⓒ

To evaluate f(a),f(a), substitute a for x.
Simplify.

Example \(\PageIndex{20}\)

For the function \(f(x)=3x^2−2x+1\), evaluate the function.

1. \(f(3)\)
2. \(f(−1)\)
3. \(f(t)\)

Answer

ⓐ \(f(3)=22\) ⓑ \(f(−1)=6\) ⓒ \(f(t)=3t^2−2t−1\)

Example \(\PageIndex{21}\)

For the function \(f(x)=2x^2+4x−3\), evaluate the function.

1. \(f(2)\)
2. \(f(−3)\)
3. \(f(h)\)

Answer

ⓐ \((2)=13\) ⓑ \(f(−3)=3\)
ⓒ \(f(h)=2h2+4h−3\)

Example \(\PageIndex{22}\)

For the function \(g(x)=3x−5\), evaluate the function.

1. \(g(h^2)\)
2. \(g(x+2)\)
3. \(g(x)+g(2)\)

Answer

ⓐ

To evaluate \(g(h^2)\), substitute \(h^2\) for x.

ⓑ

To evaluate \(g(x+2)\), substitute \(x+2\) for x.
Simplify.

ⓒ

To evaluate \(g(x)+g(2)\), first find \(g(2)\).
Simplify.

Notice the difference between part ⓑ and ⓒ. We get \(g(x+2)=3x+1\) and \(g(x)+g(2)=3x−4\). So we see that \(g(x+2)\neq g(x)+g(2)\).

Example \(\PageIndex{23}\)

For the function \(g(x)=4x−7\), evaluate the function.

1. \(g(m^2)\)
2. \(g(x−3)\)
3. \(g(x)−g(3)\)

Answer

ⓐ \(4m^2−7\) ⓑ \(4x−19\)
ⓒ \(x−12\)

Example \(\PageIndex{24}\)

For the function \(h(x)=2x+1\), evaluate the function.

1. \(h(k^2)\)
2. \(h(x+1)\)
3. \(h(x)+h(1)\)

Answer

ⓐ \(2k^2+1\) ⓑ \(2x+3\)
ⓒ \(2x+4\)

Example \(\PageIndex{25}\)

The number of unread emails in Sylvia’s account is 75. This number grows by 10 unread emails a day. The function \(N(t)=75+10t\) represents the relation between the number of emails, N, and the time, t, measured in days.

1. Determine the independent and dependent variable.
2. Find \(N(5)\). Explain what this result means.

Answer

ⓐ The number of unread emails is a function of the number of days. The number of unread emails, N, depends on the number of days, t. Therefore, the variable N, is the dependent variable and the variable tt is the independent variable.

ⓑ Find \(N(5)\). Explain what this result means.

Substitute in t=5.t=5.
Simplify.

Since 5 is the number of days, \(N(5)\), is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in the account.

Example \(\PageIndex{26}\)

The number of unread emails in Bryan’s account is 100. This number grows by 15 unread emails a day. The function \(N(t)=100+15t\) represents the relation between the number of emails, N, and the time, t, measured in days.

1. Determine the independent and dependent variable.
2. Find \(N(7)]\). Explain what this result means.

Answer

ⓐ t IND; N DEP ⓑ 205; the number of unread emails in Bryan’s account on the seventh day.

Example \(\PageIndex{27}\)

The number of unread emails in Anthony’s account is 110. This number grows by 25 unread emails a day. The function \(N(t)=110+25t\) represents the relation between the number of emails, N, and the time, t, measured in days.

1. Determine the independent and dependent variable.
2. Find \(N(14)\). Explain what this result means.

Answer

ⓐ t IND; N DEP ⓑ 460; the number of unread emails in Anthony’s account on the fourteenth day