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Study Guides > MATH 0123

Characteristics of Functions and Their Graphs

Learning Outcomes

  • Determine whether a relation represents a function.
  • Find function values.
  • Determine whether a function is one-to-one.
  • Use the vertical line test to identify functions.
  • Graph the functions in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Characteristics of Functions

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain of the relation and the set of the second components of each ordered pair is called the range of the relation. 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 the first.

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

The domain is {1,2,3,4,5}\left\{1,2,3,4,5\right\}. The range is {2,4,6,8,10}\left\{2,4,6,8,10\right\}. Note the values in the domain are also known as an input values, or values of the independent variable, and are often labeled with the lowercase letter xx. Values in the range are also known as an output values, or values of the dependent variable, and are often labeled with the lowercase letter yy. A function ff is a relation that assigns a single value in the range to each value in the domain. In other words, no xx-values are used more than once. 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}\left\{1,2,3,4,5\right\}, is paired with exactly one element in the range, {2,4,6,8,10}\left\{2,4,6,8,10\right\}. 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

{(odd,1),(even,2),(odd,3),(even,4),(odd,5)}\left\{\left(\text{odd},1\right),\left(\text{even},2\right),\left(\text{odd},3\right),\left(\text{even},4\right),\left(\text{odd},5\right)\right\}

Notice that each element in the domain, {even,odd}\left\{\text{even,}\text{odd}\right\} is not paired with exactly one element in the range, {1,2,3,4,5}\left\{1,2,3,4,5\right\}. For example, the term "odd" corresponds to three values from the domain, {1,3,5}\left\{1,3,5\right\} and the term "even" corresponds to two values from the range, {2,4}\left\{2,4\right\}. This violates the definition of a function, so this relation is not a function. This image compares relations that are functions and not functions.
Three relations that demonstrate what constitute a function. (a) This relationship is a function because each input is associated with a single output. Note that input qq and rr both give output nn. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input qq is associated with two different outputs.

A General Note: FunctionS

A function is a relation in which each possible input value leads to exactly one output value. We say "the output is a function of the input." The input values make up the domain, and the output values make up the range.

How To: Given a relationship between two quantities, determine whether the relationship is a function.

  1. Identify the input values.
  2. Identify the output values.
  3. If each input value leads to only one output value, the relationship is a function. If any input value leads to two or more outputs, the relationship is not a function.

Example: Determining If Menu Price Lists Are Functions

The coffee shop menu consists of items and their prices.
  1. Is price a function of the item?
  2. Is the item a function of the price?
A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.

Answer:

  1. Let’s begin by considering the input as the items on the menu. The output values are then the prices.A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.Each item on the menu has only one price, so the price is a function of the item.
  2. 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.Association of the prices to the donuts.Therefore, the item is a not a function of price.

Example: Determining If Class Grade Rules Are Functions

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? The table below 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

Answer: 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.

https://youtu.be/zT69oxcMhPw

Try It

The table below 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?

Answer:

  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.)

Try It

Given the function g(m)=m4g\left(m\right)=\sqrt{m - 4}, evaluate g(5)g\left(5\right).

Answer: g(5)=54=1g\left(5\right)=\sqrt{5- 4}=1

https://youtu.be/GLOmTED1UwA

Example: Solving Functions

Given the function h(p)=p2+2ph\left(p\right)={p}^{2}+2p, solve for h(p)=3h\left(p\right)=3.

Answer:

\begin{align}&h\left(p\right)=3\\ &{p}^{2}+2p=3 &&\text{Substitute the original function }h\left(p\right)={p}^{2}+2p. \\ &{p}^{2}+2p - 3=0 &&\text{Subtract 3 from each side}. \\ &\left(p+3\text{)(}p - 1\right)=0 &&\text{Factor}. \end{align}

If (p+3)(p1)=0\left(p+3\right)\left(p - 1\right)=0, either (p+3)=0\left(p+3\right)=0 or (p1)=0\left(p - 1\right)=0 (or both of them equal 0). We will set each factor equal to 0 and solve for pp in each case.

\begin{align}&p+3=0, &&p=-3 \\ &p - 1=0, &&p=1\hfill \end{align}

This gives us two solutions. The output h(p)=3h\left(p\right)=3 when the input is either p=1p=1 or p=3p=-3. Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24). We can also verify by graphing as in Figure 5. The graph verifies that h(1)=h(3)=3h\left(1\right)=h\left(-3\right)=3 and h(4)=24h\left(4\right)=24.

https://www.youtube.com/watch?v=NTmgEF_nZSc

Try It

Given the function g(m)=m4g\left(m\right)=\sqrt{m - 4}, solve g(m)=2g\left(m\right)=2.

Answer: m=8m=8

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation 2n+6p=122n+6p=12 expresses a functional relationship between nn and pp. We can rewrite it to decide if pp is a function of nn.

How To: Given a function in equation form, write its algebraic formula.

  1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
  2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example: Finding an Equation of a Function

Express the relationship 2n+6p=122n+6p=12 as a function p=f(n)p=f\left(n\right), if possible.

Answer: To express the relationship in this form, we need to be able to write the relationship where pp is a function of nn, which means writing it as p=p= expression involving nn.

\begin{align}&2n+6p=12\\[1mm] &6p=12 - 2n &&\text{Subtract }2n\text{ from both sides}. \\[1mm] &p=\frac{12 - 2n}{6} &&\text{Divide both sides by 6 and simplify}. \\[1mm] &p=\frac{12}{6}-\frac{2n}{6} \\[1mm] &p=2-\frac{1}{3}n \end{align}

Therefore, pp as a function of nn is written as

p=f(n)=213np=f\left(n\right)=2-\frac{1}{3}n

Analysis of the Solution

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

https://www.youtube.com/watch?v=lHTLjfPpFyQ

Example: Expressing the Equation of a Circle as a Function

Does the equation x2+y2=1{x}^{2}+{y}^{2}=1 represent a function with xx as input and yy as output? If so, express the relationship as a function y=f(x)y=f\left(x\right).

Answer: First we subtract x2{x}^{2} from both sides.

y2=1x2{y}^{2}=1-{x}^{2}

We now try to solve for yy in this equation.

\begin{align}y&=\pm \sqrt{1-{x}^{2}} \\[1mm] &=\sqrt{1-{x}^{2}}\hspace{3mm}\text{and}\hspace{3mm}-\sqrt{1-{x}^{2}} \end{align}

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function y=f(x)y=f\left(x\right).

Try It

If x8y3=0x - 8{y}^{3}=0, express yy as a function of xx.

Answer: y=f(x)=x32y=f\left(x\right)=\cfrac{\sqrt[3]{x}}{2}

Q & A

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula? Yes, this can happen. For example, given the equation x=y+2yx=y+{2}^{y}, if we want to express yy as a function of xx, there is no simple algebraic formula involving only xx that equals yy. However, each xx does determine a unique value for yy, and there are mathematical procedures by which yy can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for yy as a function of xx, even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours. The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.
Pet Memory span in hours
Puppy 0.008
Adult dog 0.083
Cat 16
Goldfish 2160
Beta fish 3600
At times, evaluating a function in table form may be more useful than using equations. Here let us call the function PP. The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function PP at the input value of "goldfish." We would write P(goldfish)=2160P\left(\text{goldfish}\right)=2160. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function PP seems ideally suited to this function, more so than writing it in paragraph or function form.

How To: Given a function represented by a table, identify specific output and input values.

  1. Find the given input in the row (or column) of input values.
  2. Identify the corresponding output value paired with that input value.
  3. Find the given output values in the row (or column) of output values, noting every time that output value appears.
  4. Identify the input value(s) corresponding to the given output value.

Example: Evaluating and Solving a Tabular Function

Using the table below,
  1. Evaluate g(3)g\left(3\right).
  2. Solve g(n)=6g\left(n\right)=6.
nn 1 2 3 4 5
g(n)g(n) 8 6 7 6 8

Answer:

  • Evaluating g(3)g\left(3\right) means determining the output value of the function gg for the input value of n=3n=3. The table output value corresponding to n=3n=3 is 7, so g(3)=7g\left(3\right)=7.
  • Solving g(n)=6g\left(n\right)=6 means identifying the input values, nn, that produce an output value of 6. The table below shows two solutions: n=2n=2 and n=4n=4.
nn 1 2 3 4 5
g(n)g(n) 8 6 7 6 8
When we input 2 into the function gg, our output is 6. When we input 4 into the function gg, our output is also 6.

Try It

Using the table from the previous example, evaluate g(1)g\left(1\right) .

Answer: g(1)=8g\left(1\right)=8

Finding Function Values from a Graph

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: Reading Function Values from a Graph

Given the graph below,
  1. Evaluate f(2)f\left(2\right).
  2. Solve f(x)=4f\left(x\right)=4.
Graph of a positive parabola centered at (1, 0).

Answer:

  1. To evaluate f(2)f\left(2\right), locate the point on the curve where x=2x=2, then read the yy-coordinate of that point. The point has coordinates (2,1)\left(2,1\right), so f(2)=1f\left(2\right)=1.Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.
  2. To solve f(x)=4f\left(x\right)=4, we find the output value 44 on the vertical axis. Moving horizontally along the line y=4y=4, we locate two points of the curve with output value 4:4: (1,4)\left(-1,4\right) and (3,4)\left(3,4\right). These points represent the two solutions to f(x)=4:f\left(x\right)=4: x=1x=-1 or x=3x=3. This means f(1)=4f\left(-1\right)=4 and f(3)=4f\left(3\right)=4, or when the input is 1-1 or 3,\text{3,} the output is 4.\text{4}\text{.} See the graph below.Graph of an upward-facing parabola with a vertex at (0,1) and labeled points at (-1, 4) and (3,4). A line at y = 4 intersects the parabola at the labeled points.

Try It

Using the graph, solve f(x)=1f\left(x\right)=1. Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.

Answer: x=0[/latex]or[latex]x=2x=0[/latex] or [latex]x=2

Identify Functions Using Graphs

As we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis. The most common graphs name the input value xx and the output value yy, and we say yy is a function of xx, or y=f(x)y=f\left(x\right) when the function is named ff. The graph of the function is the set of all points (x,y)\left(x,y\right) in the plane that satisfies the equation y=f(x)y=f\left(x\right). 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 the graph below tell us that f(0)=2f\left(0\right)=2 and f(6)=1f\left(6\right)=1. However, the set of all points (x,y)\left(x,y\right) satisfying y=f(x)y=f\left(x\right) is a curve. The curve shown includes (0,2)\left(0,2\right) and (6,1)\left(6,1\right) because the curve passes through those points. Graph of a polynomial. The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular xx value. The yy value of a point where a vertical line intersects a graph represents an output for that input xx value. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because that xx value has more than one output. A function has only one output value for each input value. Three graphs visually showing what is and is not a function.

How To: Given a graph, use the vertical line test to determine if the graph represents a function.

  1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
  2. If there is any such line, the graph does not represent a function.
  3. If no vertical line can intersect the curve more than once, the graph does represent a function.

Example: Applying the Vertical Line Test

Which of the graphs represent(s) a function y=f(x)?y=f\left(x\right)? Graph of a polynomial.

Answer: 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 the graph above. 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. Graph of a circle.

https://youtu.be/5Z8DaZPJLKY

Try It

Does the graph below represent a function? Graph of absolute value function.

Answer: Yes.

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. A horizontal line includes all points with a particular yy value. The xx value of a point where a vertical line intersects a function represents the input for that output yy value. If we can draw any horizontal line that intersects a graph more than once, then the graph does not represent a one-to-one function because that yy value has more than one input.

How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

  1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
  2. If there is any such line, the function is not one-to-one.
  3. If no horizontal line can intersect the curve more than once, the function is one-to-one.

Example: Applying the Horizontal Line Test

Consider the functions (a), and (b)shown in the graphs below. Graph of a polynomial. Are either of the functions one-to-one?

Answer: The function in (a) is not one-to-one. The horizontal line shown below 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.

https://youtu.be/tbSGdcSN8RE

Identifying Basic Toolkit Functions

In this text, we explore 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 xx as the input variable and y=f(x)y=f\left(x\right) 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 below.
Toolkit Functions
Name Function Graph
Constant f(x)=cf\left(x\right)=c, where cc is a constant Graph of a constant function.
Identity f(x)=xf\left(x\right)=x Graph of a straight line.
Absolute value f(x)=xf\left(x\right)=|x| Graph of absolute function.
Quadratic f(x)=x2f\left(x\right)={x}^{2} Graph of a parabola.
Cubic f(x)=x3f\left(x\right)={x}^{3} Graph of f(x) = x^3.
Reciprocal/ Rational f(x)=1xf\left(x\right)=\frac{1}{x} Graph of f(x)=1/x.
Reciprocal / Rational squared f(x)=1x2f\left(x\right)=\frac{1}{{x}^{2}} Graph of f(x)=1/x^2.
Square root f(x)=xf\left(x\right)=\sqrt{x} Graph of f(x)=sqrt(x).
Cube root f(x)=x3f\left(x\right)=\sqrt[3]{x} Graph of f(x)=x^(1/3).
 

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.
  • Function notation is a shorthand method for relating the input to the output in the form y=f(x)y=f\left(x\right).
  • In table form, a function can be represented by rows or columns that relate to input and output values.
  • 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.
  • To solve for a specific function value, we determine the input values that yield the specific output value.
  • An algebraic form of a function can be written from an equation.
  • Input and output values of a function can be identified from a table.
  • Relating input values to output values on a graph is another way to evaluate a function.
  • A function is one-to-one if each output value corresponds to only one input value.
  • A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
  • A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.

Glossary

dependent variable
an output variable
domain
the set of all possible input values for a relation
function
a relation in which each input value yields a unique output value
horizontal line test
a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once
independent variable
an input variable
input
each object or value in a domain that relates to another object or value by a relationship known as a function
one-to-one function
a function for which each value of the output is associated with a unique input value
output
each object or value in the range that is produced when an input value is entered into a function
range
the set of output values that result from the input values in a relation
relation
a set of ordered pairs
vertical line test
a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once

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