Definition of Function.
A function is a relationship between two variables for which a unique value of the output variable can be determined from a value of the input variable.
Section 0.1 Functions
Subsection 0.1.1 The Definition of Function
We often want to predict values of one variable from the values of a related variable. For a more extensive treatment of functions we refer the reader to PreCalculus at Nebraska: Introduction to Functions
2
mathbooks.unl.edu/PreCalculus/introfunctions.htmlWhat distinguishes functions from other variable relationships? The definition of a function calls for a unique value that is, exactly one value of the output variable corresponding to each value of the input variable. This property makes functions useful in applications because they can often be used to make predictions.
A function can be described in several different ways. In the following sections, we consider examples of functions defined by tables, by graphs, and by equations.
Subsection 0.1.2 Functions Defined by Tables
When we use a table to describe a function, the first variable in the table (the left column of a vertical table or the top row of a horizontal table) is the input variable, and the second variable is the output. We say that the output variable is a function of the input.
Example 0.1.
-
Table 0.2 shows data on sales compiled over several years by the accounting office for Eau Claire Auto Parts, a division of Major Motors. In this example, the year is the input variable, and total sales is the output. We say that total sales, \(S\text{,}\) is a function of \(t\text{.}\)
Table 0.2. Year \((t)\) Total sales \((S)\) 2000 $612,000 2001 $663,000 2002 $692,000 2003 $749,000 2004 $904,000 -
Table 0.3 gives the cost of sending printed material by first-class mail in 2016.
Table 0.3. Weight in ounces \((w)\) Postage \((P)\) \(0 \lt w \le 1 \) $0.47 \(1 \lt w \le 2 \) $0.68 \(2 \lt w \le 3 \) $0.89 \(3 \lt w \le 4 \) $1.10 \(4 \lt w \le 5 \) $1.31 \(5 \lt w \le 6 \) $1.52 \(6 \lt w \le 7 \) $1.73 If we know the weight of the article being shipped, we can determine the required postage from Table 0.3. For instance, a catalog weighing 4.5 ounces would require $1.31 in postage. In this example, \(w\) is the input variable and \(p\) is the output variable. We say that \(p\) is a function of \(w\text{.}\) -
Table 0.4 records the age and cholesterol count for 20 patients tested in a hospital survey.
Table 0.4. Age Cholesterol count Age Cholesterol count 53 217 \(51\) \(209\) 48 232 53 241 55 198 49 186 56 238 \(51\) \(216\) \(51\) \(227\) 57 208 52 264 52 248 53 195 50 214 47 203 56 271 48 212 53 193 50 234 48 172 According to these data, cholesterol count is not a function of age, because several patients who are the same age have different cholesterol levels. For example, three different patients are 51 years old but have cholesterol counts of 227, 209, and 216, respectively. Thus, we cannot determine a unique value of the output variable (cholesterol count) from the value of the input variable (age). Other factors besides age must influence a persons cholesterol count.
Subsection 0.1.3 Functions Defined by Graphs
A graph may also be used to define one variable as a function of another. The input variable is displayed on the horizontal axis, and the output variable on the vertical axis.
Example 0.5.
Figure 0.6 shows the number of hours, \(H\text{,}\) that the sun is above the horizon in Peoria, Illinois, on day \(t\text{,}\) where January 1 corresponds to \(t = 0\text{.}\)
- The input variable, \(t\text{,}\) appears on the horizontal axis. The number of daylight hours, \(H\text{,}\) is a function of the date. The output variable appears on the vertical axis.
- The point on the curve where \(t = 150\) has \(H \approx 14.1\text{,}\) so Peoria gets about 14.1 hours of daylight when \(t = 150\text{,}\) which is at the end of May.
- \(H = 12\) at the two points where \(t \approx 85\) (in late March) and \(t \approx 270\) (late September).
- The maximum value of 14.4 hours occurs on the longest day of the year, when \(t \approx 170\text{,}\) about three weeks into June. The minimum of 9.6 hours occurs on the shortest day, when \(t \approx 355\text{,}\) about three weeks into December.
Peoria sunlight hours
We have a method of quickly determining if a relationship is a function once we have a graph of the relationship.
The Vertical Line Test.
A graph represents a function if and only if every vertical line intersects the graph in at most one point.
Example 0.8.
Use the vertical line test to decide which of the graphs in Figure 0.9 represent functions.
vertical line test
Solution.
- Graph (a) represents a function, because it passes the vertical line test.
- Graph (b) is not the graph of a function, because the vertical line at (for example) \(x = 1\) intersects the graph at two points.
- For graph (c), notice the break in the curve at \(x = 2\text{:}\) The solid dot at \((2, 1)\) is the only point on the graph with \(x = 2\text{;}\) the open circle at \((2, 3)\) indicates that \((2, 3)\) is not a point on the graph. Thus, graph (c) is a function, with \(f(2) = 1\text{.}\)
Subsection 0.1.4 Functions Defined by Equations
Example 0.10 illustrates a function defined by an equation.
Example 0.10.
As of 2016, One World Trade Center in New York City is the nations tallest building, at 1776 feet. If an algebra book is dropped from the top of One World Trade Center, its height above the ground after \(t\) seconds is given by the equation
\begin{equation*}
h =
1776 - 16t^2
\end{equation*}
Thus, after \(1\) second the books height is
\begin{equation*}
h = 1776 -
16(1)^2 = 1760 \text{ feet}
\end{equation*}
After \(2\) seconds its height is
\begin{equation*}
h = 1776 -
16(2)^2 = 1712 \text{ feet}
\end{equation*}
For this function, \(t\) is the input variable and \(h\) is the output variable. For any value of \(t\text{,}\) a unique value of \(h\) can be determined from the equation for \(h\text{.}\) We say that \(h\) is a function of \(t\text{.}\)
Subsection 0.1.5 Function Notation
There is a convenient notation for discussing functions. First, we choose a letter, such as \(f\text{,}\) \(g\text{,}\) or \(h\) (or \(F\text{,}\) \(G\text{,}\) or \(H\)), to name a particular function. (We can use any letter, but these are the most common choices.) For instance, the height, \(h\text{,}\) of a falling textbook is a function of the elapsed time, \(t\text{.}\) We might call this function \(f\text{.}\) In other words, \(f\) is the name of the relationship between the variables \(h\) and \(t\text{.}\) We write
\begin{equation*}
h = f (t)
\end{equation*}
which means "\(h\) is a function of \(t\text{,}\) and \(f\) is the name of the function."
Example 0.11.
In Example 0.10, the height of an algebra book dropped from the top of One World Trade Center is given by the equation
\begin{equation*}
h =
1776 - 16t^2
\end{equation*}
We see that
| when \(t=1\) | \(h=1760\) | |
| when \(t=2\) | \(h=1712\) |
Using function notation, these relationships can be expressed more concisely as
| \(f(1)=1760\) | and | \(f(2)=1712\) |
which we read as "\(f\) of 1 equals 1760" and "\(f\) of 2 equals 1712." The values for the input variable, \(t\text{,}\) appear inside the parentheses, and the values for the output variable, \(h\text{,}\) appear on the other side of the equation.
Remember that when we write \(y = f(x)\text{,}\) the symbol \(f(x)\) is just another name for the output variable.
Subsection 0.1.6 Evaluating a Function
Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
Example 0.13.
Let \(g\) be the name of the postage function defined by Table 0.3. Find \(g(1)\text{,}\) \(g(3)\text{,}\) and \(g(6.75\)).
Solution.
According to the table,
| when \(w=1\text{,}\) | \(p=0.47\) | so | \(g(1)=0.47\) | |
| when \(w=3\text{,}\) | \(p=0.89\) | so | \(g(3)=0.89\) | |
| when \(w=6.75\text{,}\) | \(p=1.73\) | so | \(g(6.75)=1.73\) |
Thus, a letter weighing 1 ounce costs $0.47 to mail, a letter weighing 3 ounces costs $0.89, and a letter weighing 6.75 ounces costs $ 1.73.
If a function is described by an equation, we simply substitute the given input value into the equation to find the corresponding output, or function value.
Example 0.14.
The function \(H\) is defined by \(H=f(s) = \dfrac{\sqrt{s+3}}{s}\text{.}\) Evaluate the function at the following values.
- \(\displaystyle s=6\)
- \(\displaystyle s=-1\)
Solution.
- \(f(6)=\dfrac{\sqrt{(6)+3}}{(6)}= \dfrac{\sqrt{9}}{6}=\dfrac{3}{6}=\dfrac{1}{2}\text{.}\) Thus, \(f(6)=\dfrac{1}{2}\text{.}\)
- \(f(-1)=\dfrac{\sqrt{(-1)+3}}{(-1)}= \dfrac{\sqrt{2}}{-1}=-\sqrt{2}\text{.}\) Thus, \(f(-1)=-\sqrt{2}\) .
To simplify the notation, we sometimes use the same letter for the output variable and for the name of the function.
Subsection 0.1.7 Linear Functions
Linear relationships are relationships in which the rate of change is constant.
Linear Equation.
A linear function is a function which has a constant rate of change.
Many phenomena can be modeled using linear functions \(y = f (x)\) where the equations have the form
\begin{equation*}
f
(x) = (\text{starting value}) + (\text{rate of change}) \cdot x.
\end{equation*}
The initial value, or the value of \(f(0)\text{,}\) is the vertical intercept of the graph, and the rate of change is the slope of the graph. Thus, we can write an equation of a line as
\begin{equation*}
f
(x)
= b + mx
\end{equation*}
where the constant term, \((0,b)\text{,}\) is the vertical intercept of the line, and \(m\text{,}\) the coefficient of \(x\text{,}\) is the slope of the line. This form for an equation of a line is called the slope-intercept form.
Slope-Intercept Form.
If we write an equation of a linear function in the form,
\begin{equation*}
f (x) = mx + b
\end{equation*}
then \(m\) is the slope of the line, and \((0,b)\) is the vertical intercept although we frequently refer to the intercept by the value \(b\) since the other coordinate is always \(0\text{.}\)
Example 0.15.
An icicle grows according to the formula \(H(t)=0.05t+0.12\text{,}\) where \(t\) is the time in minutes since the first measurement was take, and \(H(t)\) is the height of the icicle in centimeters.
- The slope is 0.05, which tells us that the icicle’s height grows by 0.05 cm each minute.
- The \(y\)-intercept is 0.12, which tells us that the height of the icicle was 0.12 cm at the first measurement.
Example 0.16.
Samantha owns a catering business. For any party with up to 100 guests, she charges \(\$ 2,000\text{.}\) She charges \(\$ 8\) per person for each additional guest over 100. Give a formula for the cost of having Samantha cater your party as a function of the number of guests over 200.
Solution.
A possible formula for the cost of having Samantha cater your party is \(C(g)=2000+8g\text{,}\) where \(g\) is the number of guests over 100.
The following formula provides a method for finding the value of the slope \(m\) when given two points on the line.
Two-Point Slope Formula.
The slope of the line passing through the points \(P_1 (x_1, y_1)\) and \(P_2
(x_2, y_2)\) is given by
\begin{equation*}
m = \frac{\Delta y}{\Delta x}= \frac{y_2 - y_1}{x_2 -
x_1}
\text{, }~x_2 \ne x_1.
\end{equation*}
Example 0.17.
Compute the slope of the line between the points \((6, -3)\) and \((4, 3)\) .
Solution.
Substitute the coordinates of \(Q_1\) and \(Q_2\) into the slope formula to find
\begin{equation*}
m = \frac{y_2 - y_1}{x_2 - x_1}= \frac{3 - (-3)}{4 - 6}
= \frac{6}{-2}= -3.
\end{equation*}
This value for the slope, \(-3\text{,}\) is the same value found above.
Sometimes you will be given the slope of a line and another point on that line. The following formula is helpful in that situation.
Point-Slope Form.
An equation of the line that passes through the point \((x_1, y_1)\) and has slope \(m\) is
\begin{equation*}
y- y_1= + m(x- x_1).
\end{equation*}
Example 0.18.
Find a formula for the line that has slope -2 and passes through the point \((-1,4)\text{.}\)
Solution.
Using point-slope form, we have the line \(y=4+(-2)(x-(-1))\text{.}\)
We can also write this in slope-intercept form by simplifying: \(y=2-2x\text{.}\)
Is is also useful to introduce the term \(x\)-intercept.
\(x\)-intercept.
An \(x\)-intercept for a function \(f(x)\) is the value of \(x\) such that \(f(x)=0\text{.}\)
Note 0.19.
In the equation \(f (x) = mx + b\text{,}\) we call \(m\) and \(b\) parameters. Their values are fixed for any particular linear equation; for example, in the equation \(y = 2x + 3\text{,}\) \(m = 2\) and \(b = 3\text{,}\) and the variables are \(x\) and \(y\text{.}\) By changing the values of \(m\) and \(b\text{,}\) we can write the equation for any line except a vertical line (see Figure 0.20). The collection of all linear functions \(f (x) = mx + b\) is called a two-parameter family of functions.
lines with common intercept or common slope
Subsection 0.1.8 Describing Functions
There are several terms that will be useful in describing functions. We first begin with the notion of an increasing function.
Increasing Function.
A function \(f\) is increasing if the values of \(f(x)\) increase as \(x\) increases. The graph of an increasing function climbs as we move from left to right.
Decreasing Function.
A function \(f\) is decreasing if the values of \(f(x)\) decrease as \(x\) increases. The graph of a decreasing function falls as we move from left to right.
Monotonic Function.
A function \(f(x)\) is monotonic if it increases for all \(x\) or decreases for all \(x\text{.}\)
Directly Proportional.
We say \(y\) is directly proportional to \(x\) if there is a nonzero constant \(k\) such that, \(y = kx\text{.}\) This \(k\) is called the constant of proportionality.
Inversely Proportional.
We say that \(y\) is inversely proportional to \(x\) if \(y\) is proportional to the reciprocal of \(x\text{,}\) that is, \(y = \frac{k}{x}\) for a nonzero constant \(k\text{.}\)
Subsection 0.1.9 Function Transformations
It is also useful to talk about transformations of functions. Several key facts will be useful.
Function Transformations.
- Multiplying a function by a constant, \(c\text{,}\) stretches the graph vertically (if \(c \gt 1\)) or shrinks the graph vertically (if \(0 \lt c \lt 1\))
- A negative sign (if \(c \lt 0\)) reflects the graph about the \(x\)-axis, in addition to shrinking or stretching.
- Replacing \(y\) by \((y+k)\) moves a graph up by \(k\) (down if \(k\) is negative).
- Replacing \(x\) by \((x-h)\) moves a graph to the right by \(h\) (to the left if \(h\) is negative).
Example 0.21.
Let \(f(x)=x^2\text{.}\) We will explore different transformations performed on the graph of \(f(x)\text{.}\)
- \(g(x)=2\cdot f(x)=2x^2\) is a vertical stretch of \(f(x)\) by a factor of 2.
- \(h(x)=-2\cdot f(x)=-2x^2\) is a vertical stretch of \(f(x)\) by a factor of 2 and a reflection across the \(x\)-axis.
- \(j(x)=f(x)+5=x^2+5\) is a vertical shift of \(f(x)\) up 5 units.
- \(k(x)=f(x+5)=(x+5)^2\) is a horizontal shift of \(f(x)\) left 5 units.
Order of Transformations.
Suppose that \(f(x)\) is our function we are applying transformations to. Once written in the form
\begin{equation*}
a\cdot f(b\cdot (x+h))+k
\end{equation*}
the order of transformations is:
- horizontal stretch or compress by a factor of \(|b|\) or \(|\dfrac{1}{b}|\) (if \(b\lt 0\) then also reflect about \(y\)-axis)
- shift horizontally left/ right by \(|h| \)
- vertically stretch or compress by a factor of \(|a|\) or \(|\dfrac{1}{a}|\) (if \(a\lt 0\) then also reflect about \(x\)-axis)
- shift vertically up/ down by \(|k| \)
Example 0.22.
Describe the function
\begin{equation*}
5\cdot [f(-3x-6)-1]
\end{equation*}
as a list of transformations done to \(f(x)\) in the appropriate order.
Solution.
First, let’s rewrite the function in the form given above: \begin{align*} 5\cdot [f(-3x-6)-1]\amp = 5f(-3x-6)-5 \ \ \text{we distributed in the 5}\\ \amp = 5f(-3(x+2))-5 \ \ \text{we factored out the -3} \end{align*} Now that the function is written in the desired order, we may list off the transformations in the correct order using the order discussed above:
- horizontal compress by \(3\) and reflect about the \(y\)-axis
- shift horizontally left by \(2 \)
- vertically stretch by \(5\)
- shift vertically down by \(5 \)
Subsection 0.1.10 Inverse Functions
It is also useful to consider inverse functions.
Inverse Functions.
Suppose the inverse of \(f\) is a function, denoted by \(f^{-1}\text{.}\) Then
\begin{equation*}
f^{-1}(y)
= x \text{ if and only if }f(x) = y.
\end{equation*}
Example 0.23.
Suppose \(g\) is the inverse function for \(f\text{,}\) and we know the following function values for \(f\text{:}\)
\begin{equation*}
f (-3) = 5, ~~ f (2) = 1, ~~ f (5) = 0.
\end{equation*}
Find \(g(5)\) and \(g(0)\text{.}\)
Solution.
We know that \(g(5) = -3\) because \(f (-3) = 5\text{,}\) and \(g(0) = 5\) because \(f (5) = 0\text{.}\) Tables may be helpful in visualizing the two functions, as shown below.
| \(y=f(x)\) | |
| \(x\) | \(y\) |
| \(-3\) | \(5\) |
| \(2\) | \(1\) |
| \(5\) | \(0\) |
Interchange the columns
| \(x=g(y)\) | |
| \(y\) | \(x\) |
| \(5\) | \(-3\) |
| \(1\) | \(2\) |
| \(0\) | \(5\) |
For the function \(f\text{,}\) the input variable is \(x\) and the output variable is \(y\text{.}\) For the inverse function \(g\text{,}\) the roles of the variables are interchanged: \(y\) is now the input and \(x\) is the output.
Functions and Inverse Functions.
Suppose \(f^{-1}\) is the inverse function for \(f\text{.}\) Then
\begin{equation*}
f^{-1}\left(f(x)\right)
= x\text{ and }f\left(f^{-1}( y)\right) = y
\end{equation*}
as long as \(x\) is in the domain of \(f\text{,}\) and \(y\) is in the domain of \(f^{-1}\text{.}\)
We also have a method of quickly determining if a function is invertible once we have a graph of the function.
Horizontal Line Test.
If no horizontal line intersects the graph of a function more than once, then its inverse is also a function.
Horizontal line test
Horizontal line test
Example 0.26.
Which of the functions in Figure 0.27 are invertible?
graphs for vertical line test
Solution.
In each case, apply the horizontal line test to determine whether the function is invertible. Because no horizontal line intersects their graphs more than once, the functions pictured in Figures 0.27(a) and (c) are invertible. The functions in Figures 0.27(b) and (d) are not invertibe.
We have been talking about how to tell if the inverse of a function is also a function, but in practice this is not the language typicaly used. Usually we ask this same question in the form "Is the function invertible?" The following definition explains this relationship:
Invertible Function.
If \(y=f(x) \) is a function such that its inverse, \(x=f^{-1}(y)\text{,}\) is also a function then we say that \(f(x) \) is an invertible function.
The inverse function \(f^{-1}\) undoes the effect of the function \(f\text{.}\) The function \(f(t) = 6 + 2t\) multiplies the input by \(2\) and then adds \(6\) to the result. The inverse function \(f^{-1}(H) = \dfrac{H -6}{2}\) undoes those operations in reverse order: It subtracts \(6\) from the input and then divides the result by \(2\text{.}\)
If we apply the function \(f\) to a given input value and then apply the function \(f^{-1}\) to the output from \(f\text{,}\) the end result will be the original input value. For example, if we choose \(t = 5\) as an input value, we find that \begin{align*} f(5)\amp= 6 + 2(5) = 16\amp\amp\text{ Multiply by 2, then add 6.}\\ \text{and } f^{-1}(16) \amp = \frac{16 - 6}{2} = 5.\amp\amp\text{Subtract 6, then divide by 2.} \end{align*}
function and inverse diagram
We return to the original input value, \(5\text{.}\)
Subsection 0.1.11 Supplemental Videos
- Functions
3
unl.yuja.com/V/Video?v=7114212&node=34303239&a=39473130&autoplay=1 - Describing and Transforming Functions
4
unl.yuja.com/V/Video?v=7114211&node=34303284&a=49202872&autoplay=1 - Compositions and Inverses
5
unl.yuja.com/V/Video?v=7114210&node=34303243&a=143114257&autoplay=1
Exercises 0.1.12 Exercises
1. Slope and Intercept.
Determine the slope and the \(y\)-intercept of the line \(6 x=3 y+11\text{.}\)
Slope=
\(y\)-intercept=
2. Graphs of Linear Equations.
Match the graphs below with the equations following. (Note that the \(x\) and \(y\) scales may be unequal.)
I.  |
II.  |
III.  |
IV.  |
V.  |
VI.  |
(For each, enter the correct roman numeral, e.g., II, without any punctuation.)
| (a) \(y = -2.9 - 2 x\text{:}\) | (b) \(y = 2.2 + 1.8 x\text{:}\) |
| (c) \(y = -2.6 x\text{:}\) | (d) \(y = 2.3 x\text{:}\) |
| (e) \(y = 2.8 - 1.2 x\text{:}\) | (f) \(y = -2.4 + 2.1 x\text{:}\) |
3. Proportionality.
Write an equation that represents the following function. The strength, \(S\text{,}\) of a beam is proportional to the square of its thickness, \(h\text{.}\)
Use the letter k for any constant of proportionality in your answer.
S( ) =
4. Finding Lines.
Find the equation of the line that passes through the \((x,y)\) points \((-3 ,4 )\) and \((3 ,2 )\text{.}\)
\(y=\)
5. Transformations.
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6. Matching Transformations.
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