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Functions and Function Notation

Determine whether a relation represents a function.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

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

The domain is $\{1,\ 2,\ 3,\ 4,\ 5\}.$ The range is $\{2,\ 4,\ 6,\ 8,\ 10\}.$

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter $x.$ Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter $y.$

A function $f$ is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, $\{1,\ 2,\ 3,\ 4,\ 5\},$ is paired with exactly one element in the range, $\{2,\ 4,\ 6,\ 8,\ 10\}.$

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

\[\{(\text{odd},\ 1),\ (\text{even},\ 2),\ (\text{odd},\ 3),\ (\text{even},\ 4),\ (\text{odd},\ 5)\}\]

Notice that each element in the domain, $\{\text{even,}\ \text{odd}\}$ is not paired with exactly one element in the range, $\{1,\ 2,\ 3,\ 4,\ 5\}.$ For example, the term “odd” corresponds to three values from the range, $\{1,\ 3,\ 5\}$ and the term “even” corresponds to two values from the range, $\{2,\ 4\}.$ This violates the definition of a function, so this relation is not a function.

compares relations that are functions and not functions.

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

Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example

Try it.

Given the graph in ,

ⓐ Evaluate $f(2).$
ⓑ Solve $f(x)=4.$

Solution

ⓐ To evaluate $f(2),$ locate the point on the curve where $x=2,$ then read the y-coordinate of that point. The point has coordinates $(2,1),$ so $f(2)=1.$ See .
ⓑ To solve $f(x)=4,$ we find the output value $4$ on the vertical axis. Moving horizontally along the line $y=4,$ we locate two points of the curve with output value $4:$ $(-1,4)$ and $(3,4).$ These points represent the two solutions to $f(x)=4:$ $-1$ or $3.$ This means $f(-1)=4$ and $f(3)=4,$ or when the input is $-1$ or $\text{3,}$ the output is $\text{4}\text{.}$ See .

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

Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in .

Letter grade	Grade point average
A	4.0
B	3.0
C	2.0
D	1.0

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b). The function in part (a) shows a relationship that is not a one-to-one function because inputs $q$ and $r$ both give output $n.$ The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

Example

Try it.

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

Solution

A circle of radius $r$ has a unique area measure given by $A=\pi {r}^{2},$ so for any input, $r,$ there is only one output, $A.$ The area is a function of radius $r.$

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure $A$ is given by the formula $A=\pi {r}^{2}.$ Because areas and radii are positive numbers, there is exactly one solution: $\sqrt{\frac{A}{\pi }}.$ So the area of a circle is a one-to-one function of the circle’s radius.

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

Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value $x$ and the output value $y,$ and we say $y$ is a function of $x,$ or $y=f(x)$ when the function is named $f.$ The graph of the function is the set of all points $(x,y)$ in the plane that satisfies the equation $y=f(x).$ If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in tell us that $f(0)=2$ and $f(6)=1.$ However, the set of all points $(x,y)$ satisfying $y=f(x)$ is a curve. The curve shown includes $(0,2)$ and $(6,1)$ because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See .

Example

Try it.

Which of the graphs in represent(s) a function $y=f(x)?$

Solution

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of . From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point, as shown in .

Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Example

Try it.

Consider the functions shown in (a) and (b). Are either of the functions one-to-one?

Solution

The function in (a) is not one-to-one. The horizontal line shown in intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use $x$ as the input variable and $y=f(x)$ as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in .

Toolkit Functions
Name	Function	Graph
Constant	$f(x)=c,$ where $c$ is a constant
Identity	$f(x)=x$
Absolute value	$f(x)=\|x\|$
Quadratic	$f(x)={x}^{2}$
Cubic	$f(x)={x}^{3}$
Reciprocal	$f(x)=\frac{1}{x}$
Reciprocal squared	$f(x)=\frac{1}{{x}^{2}}$
Square root	$f(x)=\sqrt{x}$
Cube root	$f(x)=\sqrt[3]{x}$

Key Equations

Constant function	$f(x)=c,$ where $c$ is a constant
Identity function	$f(x)=x$
Absolute value function	$f(x)=\|x\|$
Quadratic function	$f(x)={x}^{2}$
Cubic function	$f(x)={x}^{3}$
Reciprocal function	$f(x)=\frac{1}{x}$
Reciprocal squared function	$f(x)=\frac{1}{{x}^{2}}$
Square root function	$f(x)=\sqrt{x}$
Cube root function	$f(x)=\sqrt[3]{x}$

Key Concepts

A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See and .
Function notation is a shorthand method for relating the input to the output in the form $y=f(x).$ See and .
In tabular form, a function can be represented by rows or columns that relate to input and output values. See .
To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See and .
To solve for a specific function value, we determine the input values that yield the specific output value. See .
An algebraic form of a function can be written from an equation. See and .
Input and output values of a function can be identified from a table. See .
Relating input values to output values on a graph is another way to evaluate a function. See .
A function is one-to-one if each output value corresponds to only one input value. See .
A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See .
The graph of a one-to-one function passes the horizontal line test. See .

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. ⓐ For the function $f(x)=2x-5$
  Find $f(4)$ , $f(-6)$ , $f(0)$ , and $f(a)$ .
  Find the value of $x$ that makes $f(x)=11$
2. ⓑ Refer to the following table of values for the function $g(x)$ .
  $x$ $g(x)$
  $0$ $2$
  $1$ $5$
  $2$ $14$
  $3$ $29$
  Find $g(1)$ , $g(3)$ , $g(0)$ .
3. ⓒ
  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.
  
  Find the BMI for a man of height $5'11$ who weighs $180$ pounds. Find the weight of a man of height $5'11$ and who has a BMI of $22.3$ .
Giải đáp
1. ⓐ In part a we are working with an equation and will begin by substituting the input value for x. Then use the order of operations to evaluate.
  $f(4)=2(4)-5=8-5=3$ \[\begin{array}{l}f(-6)=2(-6)-5=-12-5=-17 \\ f(0)=2(0)-5=0-5=-5 \\ f(a)=2(a)-5=2a-5\end{array}\] Find the value of $x$ that makes $f(x)=11$ Now we are given a $y$ value and need to solve the equation for the $x$ that yielded an $f(x)=11$ . \[\begin{array}{l}f(x)=2x-5 \\ 11=2x-5 \\ 11+5=2x \\ 16=2x \\ 8=x\end{array}\]
2. ⓑ Refer to the following table of values for the function $g(x)$ .
  $x$ $g(x)$
  $0$ $2$
  $1$ $5$
  $2$ $14$
  $3$ $29$
  To find $g(1)$ , find an $x$ of $1$ in your table and read $g(x)$ at this value, $g(1)=5$
  To find $g(3)$ , find an $x$ of $3$ in your table and read $g(x)$ at this value, $g(3)=29$
  To find $g(0)$ , find an $x$ of $0$ in your table and read $g(x)$ at this value, $g(0)=2$
  To find the value of $x$ that makes $g(x)=14$ , look through the $g(x)$ column to find an output of $14$ .
  Notice $x=2$ in this row, so $x=2$ .
3. ⓒ The representation shown in part c is called a mapping. Values in the domain (weight) map to values in the range (BMI).
  The BMI for a man of height $5'11$ who weighs $180$ pounds is $25.1$ .
  The weight for a man of height $5'11$ who has a BMI of $22.3$ is $160$ .
1. ⓐ Find: $f(0)$ .
2. ⓑ Find the values for $x$ when $f(x)=0$ .
For the function $f(x)=3{x}^{2}-2x+1$ find
1. ⓐ $f(3)$
2. ⓑ $f(-2)$
3. ⓒ $f(t)$
4. ⓓ The value(s) of x that make $f(x)=1$ .
For the function $g(x)=-4+\sqrt{3x+19}$ , find the following. Make sure to give exact values.
1. ⓐ $g(-5)$
2. ⓑ $g(2)$
3. ⓒ $g(0)$
4. ⓓ The value(s) of $x$ that make $g(x)=0$ .
Use the mapping $y=G(x)$ to find the following:
1. ⓐ $G(0)=$
2. ⓑ $G(9)=$
3. ⓒ If $G(x)=9$ , then $x=$ ________
4. ⓓ Is $G(x)$ a function? Explain why or why not.
Use the mapping $y=F(x)$ to find the following:
1. ⓐ $F(-1)=$
2. ⓑ $F(4)=$
3. ⓒ $F(2)=$
4. ⓓ If $F(x)=9$ , then $x=$ ________
5. ⓔ Is $F(x)$ a function? Explain why or why not.
If $h(x)=5x-7$
Find:
1. ⓐ $h(-3)=$
2. ⓑ $h(0)=$
3. ⓒ $h(w+4)=$
4. ⓓ $h(x)=23$ , then $x=$ ________
5. ⓔ $h(x)=-15$ , then $x=$ ________
$g(t)=2|t-5|+4$
Find:
1. ⓐ $g(-3)=$
2. ⓑ $g(0)=$
3. ⓒ $g(5)=$
4. ⓓ $g(w)=$
5. ⓔ $h(x)=20$ , then $x=$ ________
1. ⓐ Use the following description to build a function called $f(x)$ . "An input value is squared, multiplied by $-2$ and added to $3$ .”
  $f(x)=$
2. ⓑ Create a table of input/output values for $f(x)$ below. Show three numerical input values and one variable input value, and the corresponding output values in your table.
  $x$ $f(x)$
The coffee shop menu, shown below, consists of items and their prices.
ⓐ Is price a function of the item?
ⓑ Is the item a function of the price?

Giải đáp

ⓐ Let’s begin by considering the input as the items on the menu. The output values are then the prices. Each item on the menu has only one price, so the price is a function of the item.
ⓑ Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See the image below.

Therefore, the item is a not a function of price.
In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? shows a possible rule for assigning grade points.
Percent grade 0–56 57–61 62–66 67–71 72–77 78–86 87–91 92–100
Grade point average 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Giải đáp

For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.
In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.
http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014. lists the five greatest baseball players of all time in order of rank.
Player Rank
Babe Ruth 1
Willie Mays 2
Ty Cobb 3
Walter Johnson 4
Hank Aaron 5
1. ⓐIs the rank a function of the player name?
2. ⓑIs the player name a function of the rank?
Giải đáp
1. ⓐyes
2. ⓑ yes (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)
Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.

Giải đáp

The number of days in a month is a function of the name of the month, so if we name the function $f,$ we write $\text{days}=f(\text{month})$ or $d=f(m).$ The name of the month is the input to a “rule” that associates a specific number (the output) with each input.
For example, $f(\text{March})=31,$ because March has 31 days. The notation $d=f(m)$ reminds us that the number of days, $d$ (the output), is dependent on the name of the month, $m$ (the input).
A function $N=f(y)$ gives the number of police officers, $N,$ in a town in year $y.$ What does $f(2005)=300$ represent?

Giải đáp

When we read $f(2005)=300,$ we see that the input year is 2005. The value for the output, the number of police officers $(N),$ is 300. Remember, $N=f(y).$ The statement $f(2005)=300$ tells us that in the year 2005 there were 300 police officers in the town.
Use function notation to express the weight of a pig in pounds as a function of its age in days $d\text{.}$

Giải đáp

$w=f(d)$
Which table, , , or , represents a function (if any)?
Input Output
2 1
5 3
8 6
Input Output
–3 5
0 1
4 5
Input Output
1 0
5 2
5 4

Giải đáp

and define functions. In both, each input value corresponds to exactly one output value. does not define a function because the input value of 5 corresponds to two different output values.
When a table represents a function, corresponding input and output values can also be specified using function notation.
The function represented by can be represented by writing
\[f(2)=1,f(5)=3,\text{and }f(8)=6\]
Similarly, the statements
\[g(-3)=5,\ g(0)=1,\text{and }g(4)=5\]
represent the function in .
cannot be expressed in a similar way because it does not represent a function.
Does represent a function?
Input Output
1 10
2 100
3 1000

Giải đáp

yes
Evaluate $f(x)={x}^{2}+3x-4$ at:
1. ⓐ $2$
2. ⓑ $a$
3. ⓒ $a+h$
4. ⓓ Now evaluate $\frac{f(a+h)-f(a)}{h}$
Giải đáp
Replace the $x$ in the function with each specified value.
1. ⓐ Because the input value is a number, 2, we can use simple algebra to simplify. \[\begin{array}{l}\ f(2)={2}^{2}+3(2)-4 \\ \ \ \ \ \ \ =4+6-4 \\ \ \ \ \ \ \ =6\end{array}\]
2. ⓑ In this case, the input value is a letter so we cannot simplify the answer any further. \[f(a)={a}^{2}+3a-4\]
3. ⓒ With an input value of $a+h,$ we must use the distributive property. \[\begin{array}{l}f(a+h)={(a+h)}^{2}+3(a+h)-4 \\ \ \ \ \ \ \ \ \ \ \ ={a}^{2}+2ah+{h}^{2}+3a+3h-4\end{array}\]
4. ⓓ In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that \[f(a+h)={a}^{2}+2ah+{h}^{2}+3a+3h-4\]
  and we know that
  \[f(a)={a}^{2}+3a-4\]
  Now we combine the results and simplify.
  \[\begin{array}{llllllll}\begin{array}{l} \\ \frac{f(a+h)-f(a)}{h}=\frac{({a}^{2}+2ah+{h}^{2}+3a+3h-4)-({a}^{2}+3a-4)}{h}\end{array} \\ \begin{array}{llll}\ =\frac{2ah+{h}^{2}+3h}{h} & \\ \ =\frac{h(2a+h+3)}{h} & \begin{array}{lll}\begin{array}{ll} & \end{array} & \end{array}\text{Factor out }h. \\ \ =2a+h+3 & \begin{array}{lll}\begin{array}{ll} & \end{array} & \end{array}\text{Simplify}.\end{array}\end{array}\]
Given the function $h(p)={p}^{2}+2p,$ evaluate $h(4).$

Giải đáp

To evaluate $h(4),$ we substitute the value 4 for the input variable $p$ in the given function.
\[\begin{array}{l}\ h(p)={p}^{2}+2p \\ \ h(4)={(4)}^{2}+2(4) \\ \ =16+8 \\ \ =24\end{array}\]
Therefore, for an input of 4, we have an output of 24.
Given the function $g(m)=\sqrt{m-4},$ evaluate $g(5).$

Giải đáp

$g(5)=1$
Given the function $h(p)={p}^{2}+2p,$ solve for $h(p)=3.$

Giải đáp

\[\begin{array}{llll}\ h(p)=3 & & & \\ \ {p}^{2}+2p=3 & & & \text{Substitute the original function }h(p)={p}^{2}+2p. \\ {p}^{2}+2p-3=0 & & & \text{Subtract 3 from each side}. \\ \ (p+3\text{)(}p-1)=0 & & & \text{Factor}.\end{array}\]
If $(p+3)(p-1)=0,$ either $(p+3)=0$ or $(p-1)=0$ (or both of them equal 0). We will set each factor equal to 0 and solve for $p$ in each case.
\[\begin{array}{ll}(p+3)=0, & p=-3 \\ (p-1)=0, & p=1\end{array}\]
This gives us two solutions. The output $h(p)=3$ when the input is either $p=1$ or $p=-3.$ We can also verify by graphing as in . The graph verifies that $h(1)=h(-3)=3$ and $h(4)=24.$
Given the function $g(m)=\sqrt{m-4},$ solve $g(m)=2.$

Giải đáp

$m=8$
Express the relationship $2n+6p=12$ as a function $p=f(n),$ if possible.

Giải đáp

To express the relationship in this form, we need to be able to write the relationship where $p$ is a function of $n,$ which means writing it as $p=[\text{expression}\ \text{involving}\ n].$
\[\begin{array}{llll}2n+6p=12 & \\ \ \ \ \ \ \ 6p=12-2n & \begin{array}{lll} & & \end{array}\text{Subtract }2n\ \text{from both sides}. \\ \ \ \ \ \ \ \ \ p=\frac{12-2n}{6} & \begin{array}{lll} & & \end{array}\text{Divide both sides by 6 and simplify}. \\ \ \ \ \ \ \ \ \ p=\frac{12}{6}-\frac{2n}{6} & \\ \ \ \ \ \ \ \ \ p=2-\frac{1}{3}n & \end{array}\]
Therefore, $p$ as a function of $n$ is written as
\[p=f(n)=2-\frac{1}{3}n\]
Does the equation ${x}^{2}+{y}^{2}=1$ represent a function with $x$ as input and $y$ as output? If so, express the relationship as a function $y=f(x).$

Giải đáp

First we subtract ${x}^{2}$ from both sides.
\[{y}^{2}=1-{x}^{2}\]
We now try to solve for $y$ in this equation.
\[\begin{array}{l}y=\pm \sqrt{1-{x}^{2}} \\ \ =+\sqrt{1-{x}^{2}}\ \text{and }-\sqrt{1-{x}^{2}}\end{array}\]
We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $y=f(x).$
If $x-8{y}^{3}=0,$ express $y$ as a function of $x.$

Giải đáp

$y=f(x)=\frac{\sqrt[3]{x}}{2}$
Using ,
ⓐ Evaluate $g(3).$
ⓑ Solve $g(n)=6.$

$n$ 1 2 3 4 5
$g(n)$ 8 6 7 6 8

Giải đáp

ⓐ Evaluating $g(3)$ means determining the output value of the function $g$ for the input value of $n=3.$ The table output value corresponding to $n=3$ is 7, so $g(3)=7.$
ⓑ Solving $g(n)=6$ means identifying the input values, $n,$ that produce an output value of 6. The table below shows two solutions: $2$ and $4.$
$n$ 1 2 3 4 5
$g(n)$ 8 6 7 6 8
When we input 2 into the function $g,$ our output is 6. When we input 4 into the function $g,$ our output is also 6.
Using the table from Evaluating and Solving a Tabular Function above, evaluate $g(1).$

Giải đáp

$g(1)=8$
Given the graph in ,
ⓐ Evaluate $f(2).$
ⓑ Solve $f(x)=4.$

Giải đáp

ⓐ To evaluate $f(2),$ locate the point on the curve where $x=2,$ then read the y-coordinate of that point. The point has coordinates $(2,1),$ so $f(2)=1.$ See .
ⓑ To solve $f(x)=4,$ we find the output value $4$ on the vertical axis. Moving horizontally along the line $y=4,$ we locate two points of the curve with output value $4:$ $(-1,4)$ and $(3,4).$ These points represent the two solutions to $f(x)=4:$ $-1$ or $3.$ This means $f(-1)=4$ and $f(3)=4,$ or when the input is $-1$ or $\text{3,}$ the output is $\text{4}\text{.}$ See .
Using , solve $f(x)=1.$

Giải đáp

$x=0$ or $x=2$
Is the area of a circle a function of its radius? If yes, is the function one-to-one?

Giải đáp

A circle of radius $r$ has a unique area measure given by $A=\pi {r}^{2},$ so for any input, $r,$ there is only one output, $A.$ The area is a function of radius $r.$
If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure $A$ is given by the formula $A=\pi {r}^{2}.$ Because areas and radii are positive numbers, there is exactly one solution: $\sqrt{\frac{A}{\pi }}.$ So the area of a circle is a one-to-one function of the circle’s radius.
1. ⓐ Is a balance a function of the bank account number?
2. ⓑ Is a bank account number a function of the balance?
3. ⓒ Is a balance a one-to-one function of the bank account number?
Giải đáp
1. ⓐ yes, because each bank account has a single balance at any given time;
2. ⓑ no, because several bank account numbers may have the same balance;
3. ⓒ no, because the same output may correspond to more than one input.
Evaluate the following:
ⓐ If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
ⓑ If so, is the function one-to-one?
Giải đáp
1. ⓐ Yes, letter grade is a function of percent grade;
2. ⓑ No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.
Which of the graphs in represent(s) a function $y=f(x)?$

Giải đáp

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of . From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point, as shown in .
Does the graph in represent a function?

Giải đáp

yes
Consider the functions shown in (a) and (b). Are either of the functions one-to-one?

Giải đáp

The function in (a) is not one-to-one. The horizontal line shown in intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)
The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.
Is the graph shown in (c) one-to-one?

Giải đáp

No, because it does not pass the horizontal line test.
What is the difference between a relation and a function?

Giải đáp

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.
What is the difference between the input and the output of a function?
Why does the vertical line test tell us whether the graph of a relation represents a function?

Giải đáp

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.
How can you determine if a relation is a one-to-one function?

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\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: Functions and Function Notation

Determine whether a relation represents a function.
Find the value of a function.
Determine whether a function is one-to-one.
Use the vertical line test to identify functions.
Graph the functions listed in the library of functions.
Find the value of a function (IA 3.5.3)
Equations
Tables of input and output values

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.

Thử đi.

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

More in Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

\(x\)	\(g(x)\)
\(0\)	\(2\)
\(1\)	\(5\)
\(2\)	\(14\)
\(3\)	\(29\)

\(x\)	\(g(x)\)
\(0\)	\(2\)
\(1\)	\(5\)
\(2\)	\(14\)
\(3\)	\(29\)

Constant function	\(f(x)=c,\) where \(c\) is a constant
Identity function	\(f(x)=x\)
Absolute value function	\(f(x)=\|x\|\)
Quadratic function	\(f(x)={x}^{2}\)
Cubic function	\(f(x)={x}^{3}\)
Reciprocal function	\(f(x)=\frac{1}{x}\)
Reciprocal squared function	\(f(x)=\frac{1}{{x}^{2}}\)
Square root function	\(f(x)=\sqrt{x}\)
Cube root function	\(f(x)=\sqrt[3]{x}\)

Percent grade	0–56	57–61	62–66	67–71	72–77	78–86	87–91	92–100
Grade point average	0.0	1.0	1.5	2.0	2.5	3.0	3.5	4.0

Player	Rank
Babe Ruth	1
Willie Mays	2
Ty Cobb	3
Walter Johnson	4
Hank Aaron	5

\(x\)	\(f(x)\)

Input	Output
1	10
2	100
3	1000

\(n\)	1	2	3	4	5
\(g(n)\)	8	6	7	6	8

\(n\)	1	2	3	4	5
\(g(n)\)	8	6	7	6	8