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Relations and Functions

Find the domain and range of a relation

Find the Domain and Range of a Relation

As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.

When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.

\[\text{(Student name, Student ID \#)}\]

The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation.

There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.

Example

Try it.

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

ⓐ Find the domain of the relation.

ⓑ Find the range of the relation.

Solution

\(\ \{(1,1),(2,4),(3,9),(4,16),(5,25)\}\)

ⓐ The domain is the set of all x-values of the relation. \(\ \{1,2,3,4,5\}\)

ⓑ The range is the set of all y-values of the relation. \(\ \{1,4,9,16,25\}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Determine if a Relation is a Function

A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

The birthday example from helps us understand this definition. Every person has a birthday but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24th as their birthday and that June and Liz share August 2nd. Since each person has exactly one birthday, the relation in is a function.

The relation shown by the graph in includes the ordered pairs \((-3,-1)\) and \((-3,4).\) Is that okay in a function? No, as this is like one person having two different birthdays.

Example

Try it.

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.

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

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

Solution

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

(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: \(\{0,1,4,9\}.\)

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

In algebra, more often than not, functions will be represented by an equation. It is easiest to see if the equation is a function when it is solved for y. If each value of x results in only one value of y, then the equation defines a function.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Find the Value of a Function

It is very convenient to name a function and most often we name it f, g, h, F, G, or H. In any function, for each x-value from the domain we get a corresponding y-value in the range. For the function f, we write this range value y as \(f(x).\) This is called function notation and is read f of x or the value of f at x. In this case the parentheses does not indicate multiplication.

We call x the independent variable as it can be any value in the domain. We call y the dependent variable as its value depends on x.

Much as when you first encountered the variable x, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.

Let’s look at the equation \(y=4x-5.\) To find the value of y when \(x=2,\) we know to substitute \(x=2\) into the equation and then simplify.

Let \(x=2.\)

The value of the function at \(x=2\) is 3.

We do the same thing using function notation, the equation \(y=4x-5\) can be written as \(f(x)=4x-5.\) To find the value when \(x=2,\) we write:

Let \(x=2.\)

The value of the function at \(x=2\) is 3.

Example

Try it.

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

ⓐ \(f(3)\) ⓑ \(f(-2)\) ⓒ \(f(a)\)

Solution

ⓐ

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

ⓑ

Simplify.

ⓒ

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

Example

Try it.

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

ⓐ \(g({h}^{2})\) ⓑ \(g(x+2)\) ⓒ \(g(x)+g(2)\)

Solution

ⓐ

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 \(g(x+2)\ne g(x)+g(2).\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

  • Function Notation: For the function \(y=f(x)\)
    • f is the name of the function
    • x is the domain value
    • \(f(x)\) is the range value y corresponding to the value x
      We read \(f(x)\) as f of x or the value of f at x.
  • Independent and Dependent Variables: For the function \(y=f(x),\)
    • x is the independent variable as it can be any value in the domain
    • y is the dependent variable as its value depends on x

Relations and Functions

Find the Domain and Range of a Relation

In the following exercises, for each relation ⓐ find the domain of the relation ⓑ find the range of the relation.

Try it.

\(\{(1,4),(2,8),(3,12),(4,16),(5,20)\}\)

Solution

ⓐ {1, 2, 3, 4, 5} ⓑ {4, 8, 12, 16, 20}

Try it.

\(\{(1,-2),(2,-4),(3,-6),(4,-8),(5,-10)\}\)

Try it.

\(\{(1,7),(5,3),(7,9),(-2,-3),(-2,8)\}\)

Solution

ⓐ {1, 5, 7, −2} ⓑ {7, 3, 9, −3, 8}

Try it.

\(\{(11,3),(-2,-7),(4,-8),(4,17),(-6,9)\}\)

In the following exercises, use the mapping of the relation to ⓐ list the ordered pairs of the relation, ⓑ find the domain of the relation, and ⓒ find the range of the relation.

Try it.


Solution

ⓐ (Rebecca, January 18), (Jennifer, April 1), (John, January 18), (Hector, June 23), (Luis, February 15), (Ebony, April 7), (Raphael, November 6), (Meredith, August 19), (Karen, August 19), (Joseph, July 30)
ⓑ {Rebecca, Jennifer, John, Hector, Luis, Ebony, Raphael, Meredith, Karen, Joseph}
ⓒ {January 18, April 1, June 23, February 15, April 7, November 6, August 19, July 30}

Try it.


Try it.

For a woman of height \(5'4″\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5-24.9\) is considered healthy.

Solution

ⓐ (+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5) ⓑ {+100, 110, 120, 130, 140, 150, 160,} ⓒ {17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}

Try it.

For a man of height \(5'11''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5-24.9\) is considered healthy.

In the following exercises, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

Try it.


Solution

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

Try it.


Try it.


Solution

ⓐ (1, 4), (1, −4), (−1, 4), (−1, −4), (0, 3), (0, −3) ⓑ {−1, 0, 1} ⓒ {−4, −3, 3,4}

Try it.


Determine if a Relation is a Function

In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.

Try it.

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

Solution

ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3} ⓒ {9, 4, 1, 0}

Try it.

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

Try it.

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

Solution

ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3} ⓒ 0, 1, 8, 27}

Try it.

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

In the following exercises, use the mapping to ⓐ determine whether the relation is a function, ⓑ find the domain of the function, and ⓒ find the range of the function.

Try it.


Solution

ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3} ⓒ {0, 1, 2, 3}

Try it.


Try it.


Solution

ⓐ no ⓑ {Jenny, R and y, Dennis, Emily, Raul} ⓒ {RHern and ez@state.edu, JKim@gmail.com, Raul@gmail.com, ESmith@state.edu, DBroen@aol.com, jenny@aol.cvom, R and y@gmail.com}

Try it.


In the following exercises, determine whether each equation is a function.

Try it.


ⓐ \(2x+y=-3\)
ⓑ \(y={x}^{2}\)
ⓒ \(x+{y}^{2}=-5\)

Solution

ⓐ yes ⓑ yes ⓒ no

Try it.


ⓐ \(y=3x-5\)
ⓑ \(y={x}^{3}\)
ⓒ \(2x+{y}^{2}=4\)

Try it.


ⓐ \(y-3{x}^{3}=2\)
ⓑ \(x+{y}^{2}=3\)
ⓒ \(3x-2y=6\)

Solution

ⓐ yes ⓑ no ⓒ yes

Try it.


ⓐ \(2x-4y=8\)
ⓑ \(-4={x}^{2}-y\)
ⓒ \({y}^{2}=\text{-}x+5\)

Find the Value of a Function

In the following exercises, evaluate the function: ⓐ \(f(2)\) ⓑ \(f(-1)\) ⓒ \(f(a).\)

Try it.

\(f(x)=5x-3\)

Solution

ⓐ \(f(2)=7\) ⓑ \(f(-1)=-8\) ⓒ \(f(a)=5a-3\)

Try it.

\(f(x)=3x+4\)

Try it.

\(f(x)=-4x+2\)

Solution

ⓐ \(f(2)=-6\) ⓑ \(f(-1)=6\) ⓒ \(f(a)=-4a+2\)

Try it.

\(f(x)=-6x-3\)

Try it.

\(f(x)={x}^{2}-x+3\)

Solution

ⓐ \(f(2)=5\) ⓑ \(f(-1)=5\)
ⓒ \(f(a)={a}^{2}-a+3\)

Try it.

\(f(x)={x}^{2}+x-2\)

Try it.

\(f(x)=2{x}^{2}-x+3\)

Solution

ⓐ \(f(2)=9\) ⓑ \(f(-1)=6\)
ⓒ \(f(a)=2{a}^{2}-a+3\)

Try it.

\(f(x)=3{x}^{2}+x-2\)

In the following exercises, evaluate the function: ⓐ \(g({h}^{2})\) ⓑ \(g(x+2)\) ⓒ \(g(x)+g(2).\)

Try it.

\(g(x)=2x+1\)

Solution

ⓐ \(g({h}^{2})=2{h}^{2}+1\)
ⓑ \(g(x+2)=2x+5\)
ⓒ \(g(x)+g(2)=2x+6\)

Try it.

\(g(x)=5x-8\)

Try it.

\(g(x)=-3x-2\)

Solution

ⓐ \(g({h}^{2})=-3{h}^{2}-2\)
ⓑ \(g(x+2)=-3x-8\)
ⓒ \(g(x)+g(2)=-3x-10\)

Try it.

\(g(x)=-8x+2\)

Try it.

\(g(x)=3-x\)

Solution

ⓐ \(g({h}^{2})=3-{h}^{2}\)
ⓑ \(g(x+2)=1-x\)
ⓒ \(g(x)+g(2)=4-x\)

Try it.

\(g(x)=7-5x\)

In the following exercises, evaluate the function.

Try it.

\(f(x)=3{x}^{2}-5x;\) \(f(2)\)

Solution

2

Try it.

\(g(x)=4{x}^{2}-3x;\) \(g(3)\)

Try it.

\(F(x)=2{x}^{2}-3x+1;\)
\(F(-1)\)

Solution

6

Try it.

\(G(x)=3{x}^{2}-5x+2;\)
\(G(-2)\)

Try it.

\(h(t)=2|t-5|+4;\) \(h(-4)\)

Solution

22

Try it.

\(h(y)=3|y-1|-3;\) \(h(-4)\)

Try it.

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

Solution

4

Try it.

\(g(x)=\frac{x-2}{x+2};\) \(g(4)\)

In the following exercises, solve.

Try it.

The number of unwatched shows in Sylvia’s DVR is 85. This number grows by 20 unwatched shows per week. The function \(N(t)=85+20t\) represents the relation between the number of unwatched shows, N, and the time, t, measured in weeks.

ⓐ Determine the independent and dependent variable.

ⓑ Find \(N(4).\) Explain what this result means

Solution

ⓐ t IND; N DEP
ⓑ \(N(4)=165\) the number of unwatched shows in Sylvia’s DVR at the fourth week.

Try it.

Every day a new puzzle is downloaded into Ken’s account. Right now he has 43 puzzles in his account. The function \(N(t)=43+t\) represents the relation between the number of puzzles, N, and the time, t, measured in days.

ⓐ Determine the independent and dependent variable.

ⓑ Find \(N(30).\) Explain what this result means.

Try it.

The daily cost to the printing company to print a book is modeled by the function \(C(x)=3.25x+1500\) where C is the total daily cost in dollars and x is the number of books printed.

ⓐ Determine the independent and dependent variable.

ⓑ Find \(C(0).\) Explain what this result means.

ⓒ Find \(C(1000).\) Explain what this result means.

Solution

ⓐ x IND; C DEP
ⓑ \(N(0)=1500\) the daily cost if no books are printed
ⓒ \(N(1000)=4750\) the daily cost of printing 1000 books

Try it.

The daily cost to the manufacturing company is modeled by the function \(C(x)=7.25x+2500\) where \(C(x)\) is the total daily cost and x is the number of items manufactured.

ⓐ Determine the independent and dependent variable.

ⓑ Find \(C(0).\) Explain what this result means.

ⓒ Find \(C(1000).\) Explain what this result means.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Evaluate \(3x-5\) when \(x=-2\).
    If you missed this problem, review .

    Kuratidza mhinduro

    \(-11\)

  2. Evaluate \(2{x}^{2}-x-3\) when \(x=a.\)
    If you missed this problem, review .

    Kuratidza mhinduro

    \(2{a}^{2}-a-3\)

  3. Simplify: \(7x-1-4x+5.\)
    If you missed this problem, review .

    Kuratidza mhinduro

    \(3x+4\)

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

    ⓐ Find the domain of the relation.

    ⓑ Find the range of the relation.

    Kuratidza mhinduro

    \(\ \{(1,1),(2,4),(3,9),(4,16),(5,25)\}\)

    ⓐ The domain is the set of all x-values of the relation. \(\ \{1,2,3,4,5\}\)

    ⓑ The range is the set of all y-values of the relation. \(\ \{1,4,9,16,25\}\)

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

    ⓐ Find the domain of the relation.

    ⓑ Find the range of the relation.

    Kuratidza mhinduro

    ⓐ \(\{1,2,3,4,5\}\)
    ⓑ \(\{1,8,27,64,125\}\)

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

    ⓐ Find the domain of the relation.

    ⓑ Find the range of the relation.

    Kuratidza mhinduro

    ⓐ \(\{1,2,3,4,5\}\)
    ⓑ \(\{3,6,9,12,15\}\)

  7. Use the mapping of the relation shown to ⓐ list the ordered pairs of the relation, ⓑ find the domain of the relation, and ⓒ find the range of the relation.

    Kuratidza mhinduro

    ⓐ 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}

  8. Use the mapping of the relation shown to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

    Kuratidza mhinduro

    ⓐ (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}

  9. Use the mapping of the relation shown to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

    Kuratidza mhinduro

    ⓐ (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}

  10. Use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

    Kuratidza mhinduro

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

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

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

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

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

  11. Use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

    Kuratidza mhinduro

    ⓐ \((-3,3),(-2,2),(-1,0),\)
    \((0,-1),(2,-2),(4,-4)\)
    ⓑ \(\{-3,-2,-1,0,2,4\}\)
    ⓒ \(\{3,2,0,-1,-2,-4\}\)

  12. Use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

    Kuratidza mhinduro

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

  13. 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.

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

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

    Kuratidza mhinduro

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

    (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: \(\{0,1,4,9\}.\)

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

  14. 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.

    ⓐ \(\{(-3,-6),(-2,-4),(-1,-2),(0,0),(1,2),(2,4),(3,6)\}\)

    ⓑ \(\{(8,-4),(4,-2),(2,-1),(0,0),(2,1),(4,2),(8,4)\}\)

    Kuratidza mhinduro

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

  15. 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.

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

    ⓑ \(\{(7,-3),(-5,-4),(8,0),(0,0),(-6,4),(-2,2),(-1,3)\}\)

    Kuratidza mhinduro

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

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

    Kuratidza mhinduro

    ⓐ 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:

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

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

    Kuratidza mhinduro

    ⓐ 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}

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

    Kuratidza mhinduro

    ⓐ 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}

  19. Determine whether each equation is a function. Assume \(x\) is the independent variable.

    ⓐ \(2x+y=7\) ⓑ \(y={x}^{2}+1\) ⓒ \(x+{y}^{2}=3\)

    Kuratidza mhinduro

    ⓐ \(2x+y=7\)

    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.\(y=1\ y=-1\)

    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.

  20. Determine whether each equation is a function.

    ⓐ \(4x+y=-3\) ⓑ \(x+{y}^{2}=1\) ⓒ \(y-{x}^{2}=2\)

    Kuratidza mhinduro

    ⓐ yes ⓑ no ⓒ yes

  21. Determine whether each equation is a function.

    ⓐ \(x+{y}^{2}=4\) ⓑ \(y={x}^{2}-7\) ⓒ \(y=5x-4\)

    Kuratidza mhinduro

    ⓐ no ⓑ yes ⓒ yes

  22. For the function \(f(x)=2{x}^{2}+3x-1,\) evaluate the function.

    ⓐ \(f(3)\) ⓑ \(f(-2)\) ⓒ \(f(a)\)

    Kuratidza mhinduro

    ⓐ

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

    ⓑ

    Simplify.

    ⓒ

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

  23. For the function \(f(x)=3{x}^{2}-2x+1,\) evaluate the function.

    ⓐ \(f(3)\) ⓑ \(f(-1)\) ⓒ \(f(t)\)

    Kuratidza mhinduro

    ⓐ \(f(3)=22\) ⓑ \(f(-1)=6\) ⓒ \(f(t)=3{t}^{2}-2t+1\)

  24. For the function \(f(x)=2{x}^{2}+4x-3,\) evaluate the function.

    ⓐ \(f(2)\) ⓑ \(f(-3)\) ⓒ \(f(h)\)

    Kuratidza mhinduro

    ⓐ \((2)=13\) ⓑ \(f(-3)=3\)
    ⓒ \(f(h)=2{h}^{2}+4h-3\)

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

    ⓐ \(g({h}^{2})\) ⓑ \(g(x+2)\) ⓒ \(g(x)+g(2)\)

    Kuratidza mhinduro

    ⓐ

    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 \(g(x+2)\ne g(x)+g(2).\)

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

    ⓐ \(g({m}^{2})\) ⓑ \(g(x-3)\) ⓒ \(g(x)-g(3)\)

    Kuratidza mhinduro

    ⓐ \(4{m}^{2}-7\) ⓑ \(4x-19\)
    ⓒ \(4x-12\)

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

    ⓐ \(h({k}^{2})\) ⓑ \(h(x+1)\) ⓒ \(h(x)+h(1)\)

    Kuratidza mhinduro

    ⓐ \(2{k}^{2}+1\) ⓑ \(2x+3\)
    ⓒ \(2x+4\)

  28. 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.

    ⓐ Determine the independent and dependent variable.

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

    Kuratidza mhinduro

    ⓐ 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 \(t\) is the independent variable.

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

    Substitute in \(t=5.\)
    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.

  29. 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.

    ⓐ Determine the independent and dependent variable.

    ⓑ Find \(N(7).\) Explain what this result means.

    Kuratidza mhinduro

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

  30. 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.

    ⓐ Determine the independent and dependent variable.

    ⓑ Find \(N(14).\) Explain what this result means.

    Kuratidza mhinduro

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

  31. \(\{(1,4),(2,8),(3,12),(4,16),(5,20)\}\)

    Kuratidza mhinduro

    ⓐ {1, 2, 3, 4, 5} ⓑ {4, 8, 12, 16, 20}

  32. \(\{(1,-2),(2,-4),(3,-6),(4,-8),(5,-10)\}\)

  33. \(\{(1,7),(5,3),(7,9),(-2,-3),(-2,8)\}\)

    Kuratidza mhinduro

    ⓐ {1, 5, 7, −2} ⓑ {7, 3, 9, −3, 8}

  34. \(\{(11,3),(-2,-7),(4,-8),(4,17),(-6,9)\}\)

  35. For a woman of height \(5'4″\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5-24.9\) is considered healthy.

    Kuratidza mhinduro

    ⓐ (+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5) ⓑ {+100, 110, 120, 130, 140, 150, 160,} ⓒ {17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}

  36. For a man of height \(5'11''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5-24.9\) is considered healthy.

  37. \(\{(-3,9),(-2,4),(-1,1),\)
    \((0,0),(1,1),(2,4),(3,9)\}\)

    Kuratidza mhinduro

    ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3} ⓒ {9, 4, 1, 0}

  38. \(\{(9,-3),(4,-2),(1,-1),\)
    \((0,0),(1,1),(4,2),(9,3)\}\)

  39. \(\{(-3,27),(-2,8),(-1,1),\)
    \((0,0),(1,1),(2,8),(3,27)\}\)

    Kuratidza mhinduro

    ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3} ⓒ 0, 1, 8, 27}

  40. \(\{(-3,-27),(-2,-8),(-1,-1),\)
    \((0,0),(1,1),(2,8),(3,27)\}\)

Symbols used here

i
imaginary unit
i² = −1.
\neq
not equal
The two sides are different.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Relations and Functions

  1. Find the domain and range of a relation
  2. Determine if a relation is a function
  3. Find the value of a function

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Tarisa yako

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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