1.2.1What if the function is in pieces?

Piecewise-Defined Functions and Continuity

When working with your team to solve problems in this course, it is important to work effectively with other people. Effective math conversations are a valuable part of the learning process throughout this course. Choose a member of your team to read the Collaborative Learning Expectations out loud.

COLLABORATIVE LEARNING EXPECTATIONS

Working with other students allows you to develop new ways of thinking about mathematics, helps you to learn to communicate about math, and helps you to understand ideas better by having to explain your thinking to others. The following expectations will help you get the most out of working together.

An effective, participating team member will:

Respect the right of others to learn.
Help anyone on the team who asks—by giving hints and asking good questions, but not by giving answers right away.
Assist in creating team questions to ask the teacher.
Justify and explain ideas, instead of giving up when others do not understand.
Listen carefully to all team members and consider their responses thoroughly.
Not leave anyone behind or let anyone work ahead.
Not talk to another team.

1-11.

On the same set of axes, graph the functions $g(x) = \frac { 1 } { 2 }x + 1$ for $x < 4$ , and $h(x) = −x + 6$ for $x ≥ 4$ .

What happens to the graph at $x = 4$ ?
When we combine parts of several functions to make a single function, we call it a piecewise-defined function. Just as the graphs can be drawn on the same set of axes, the algebraic functions of $g$ and $h$ can be “pieced together” as a single function. This function can be written as $f ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { 2 } x + 1 } & { \text { for } x < 4 } \\ { - x + 6 } & { \text { for } x \geq 4 } \end{array} \right.$ . Evaluate $f(0)$ , $f(4)$ , and $f(6$ ). If using a TI calculator or Digital Graphing Calculator, follow the links to view video instructions for inputting piecewise-defined functions.

Math Notes

Intuitive Notion of Continuity

The formal definition of continuity will be given in Chapter 2. For now, we can say that a function is continuous if the graph of the function can be drawn without lifting your pencil from the paper.

The graph of a continuous function is shown at right. The equation of this piecewise-defined function is:

$f ( x ) = \left\{ \begin{array} { l l } { 9 - 2 ^ { x } } & { \text { for } x \leq 3 } \\ { \sqrt { x - 2 } } & { \text { for } x > 3 } \end{array} \right.$

The graph is continuous at $x = 3$ because $9 − 2^3 = \sqrt { 3 - 2 }$ . Though the graph is connected at $x = 3$ , notice that the domains of the individual equations do not overlap at $x = 3$ . This ensures that the piecewise-defined equation is a function.

Piecewise graph, left curve coming from upper left, passing through the point (0, comma 8), opening down stopping at the point (3, comma 1), right curve, starting at the same point (3, comma 0), opening down, continuing up & right.

When the values of a function are not connected as in problem 1-11, we say that the function is not continuous. The function $g ( x ) = \left\{ \begin{array} { l l } { 0.5 x ^ { 2 } + 1 \text { for } x < 4 } \\ { 1.5 x \text { for } x \geq 4 } \end{array} \right.$ is not continuous at $x = 4$ because $(0.5)4^2 + 1 ≠ 1.5(4)$ . This can be seen in the graph at right.

Piecewise graph, left curve coming from upper left, turning at the point (0, comma 1), stopping at the open point (4, comma 9), right ray, starting at closed point (4, comma 6), continuing up & right.

It is important to note that polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at all points in their domains.

1-12.

Examine the graph of the function $h ( x ) = \left\{ \begin{array} { l l } { 9 - 2 ^ { x } } & { \text { for } x \leq 3 } \\ { \sqrt { x - 2 } } & { \text { for } x > 3 } \end{array} \right.$ in the preceding Math Notes box.

In your own words, explain why $h$ is continuous at $x = 3$ .
Explain why $h$ can also be defined as $h ( x ) = \left\{ \begin{array} { l l } { 9 - 2 ^ { x } } & { \text { for } x < 3 } \\ { \sqrt { x - 2 } } & { \text { for } x \geq 3 } \end{array} \right.$ .
Sort the following functions into the categories listed below: $y = \sin(x)$ , $y = \ln(x)$ , $y = \sqrt { x }$ , $y = \lbrack{x}\rbrack$ (the greatest integer function), $y = x^{1/3}$ , $y = \arctan(x)$ , $y = \frac { 1 } { x }$ , $y = \sec(x)$ , $y = x^3$ , $y = e^x$ , $y = \tan(x)$ , $y = \frac { x ^ { 2 } } { x }$
Category I: Continuous for all real values of $x$ .
Category II: Continuous on its domain only.
Category III: Discontinuous on its domain.

1-13.

Explain why the two forms of set notation for the domain in the preceding Math Notes box are equivalent.

1-14.

Compare the domains of $f$ , $g$ , and $h$ . Explain what aspect of each function limits the domain.

$f(x) =\sqrt { x - 25 }$

$g(x) =\frac { 1 } { x - 25 }$

$h(x) = \log(x − 25)$

1-15.

Given $f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } } & { \text { for } x \leq 0 } \\ { 2 x + 1 } & { \text { for } x > 0 } \end{array} \right.$ and its graph at right, match each transformation to its graph. Test your ideas using the 1-15 Student eTool.

Piecewise graph, labeled f of x, left piece, upward parabola, starting in upper left, passing through the point (negative 2, comma 4), stopping at the vertex at (0, comma 0), right piece, ray starting at open point (0, comma 1), passing through the point (1, comma 3), continuing up & right.

$f ( x + 2 ) = \left\{ \begin{array} { l l } { ( x + 2 ) ^ { 2 } } & { \text { for } x \leq - 2 } \\ { 2 ( x + 2 ) + 1 } & { \text { for } x > - 2 } \end{array} \right.$
$f ( x ) + 2 = \left\{ \begin{array} { l l } { x ^ { 2 } + 2 } & { \text { for } x \leq 0 } \\ { 2 x + 3 } & { \text { for } x > 0 } \end{array} \right.$
$f ( - x ) = \left\{ \begin{array} { l l } { ( - x ) ^ { 2 } } & { \text { for } x \geq 0 } \\ { 2 ( - x ) + 1 } & { \text { for } x < 0 } \end{array} \right.$

Piecewise graph, labeled 1, left piece, ray starting at open point (0, comma 1), passing through the point (negative 1, comma 3), continuing up & left, right piece, upward parabola, starting at the vertex at (0, comma 0), passing through the point (2, comma 4), continuing up & right.

Piecewise graph, labeled 2, left piece, upward parabola, starting in upper left, passing through the point (negative 4, comma 4), stopping at the vertex at (negative 2, comma 0), right piece, ray starting at open point (negative 2, comma 1), passing through the point (negative 1, comma 3), continuing up & right.

Piecewise graph, labeled 3, left piece, upward parabola, starting in upper left, passing through the point (negative 1, comma 3), stopping at the vertex at (0, comma 2), right piece, ray starting at open point (0, comma 3), passing through the point (1, comma 5), continuing up & right.

State the domain and range of $f$ .
At what value of $x$ is the original function discontinuous?
At what value of $x$ is $y = f(x + 2)$ discontinuous? What about $y = f(x) + 2$ ? Explain the different answers.

1-16.

There is a debate among mathematicians over whether $f(x) =| x |$ should be considered a parent graph, or if it is simply a piecewise-defined function such that:

$f ( x ) = | x | = \left\{ \begin{array}{l}{ - x \text { for } x \leq 0 }\\{ x \text { for } x > 0 }\end{array} \right.$

Write $g(x) =| x - 2 |$ as a piecewise-defined function.
Write $h ( x ) = \left\{ \begin{array} { c c c } { - ( x + 5 ) } & { \text { for } } & { x \leq - 5 } \\ { x + 5 } & { \text { for } } & { x > - 5 } \end{array} \right.$ as an absolute value function.
Why do you think this issue is debatable?

1-17.

Let $f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } + 2 } & { \text { for } x < 1 } \\ { - x } & { \text { for } x \geq 1 } \end{array} \right.$ .

Sketch the graph of $y = f(x)$ .
Modify one piece of the function so that it is continuous.

1-18.

Determine values of $a$ and $b$ such that $g$ is a continuous function. Use the 1-18 Student eTool to verify your ideas.

$g ( x ) = \left\{ \begin{array} { l l } { \sqrt { x + 3 } } & { \text { for } x < 1 } \\ { a ( x - 1 ) ^ { 2 } + b } & { \text { for } 1 \leq x < 3 } \\ { - x + 2 } & { \text { for } x \geq 3 } \end{array} \right.$

1-19.

Selected values of a continuous function are shown in the table below.

$x$	$–4$	$–2$	$0$	$2$	$4$	$6$
$f(x)$	$7$	$5$	$3$	$1$	$1$	$3$

If the graph of $y = f(x)$ has one unique minimum point, where do you think this point is? Explain your thinking.
Could the graph of $y = f(x)$ be a parabola? If so, write a possible equation for $f$ . If not, explain why not.
Could the graph of $y = f(x)$ be an absolute value function? If so, write a possible equation for $f$ . If not, explain why not.
Is it possible that $f(3) = 10$ ? Explain.
Is it possible that $f$ has a vertical asymptote at $x = 3$ ? Explain.

1-20.

Let $g ( x ) = \left\{ \begin{array} { l l l } { 2 } & { \text { for } } & { 0 \leq x \leq 3 } \\ { 3 } & { \text { for } } & { 3 < x \leq 5 } \\ { 7 } & { \text { for } } & { 5 < x \leq 8 } \end{array} \right.$ . 1-20 HW eTool Homework Help ✎

Sketch the graph of $y = g(x)$ . Is this function continuous?
Shade the area between $g$ and the $x$ -axis. What is the shaded area?
g is an example of a step function. Why do you think it is called a step function?

1-21.

Given the functions below, compute the following values Homework Help ✎

$f ( x ) = \left\{ \begin{array} { c c c } { 4 - 3 x } & { \text { for } } & { x \leq 1 } \\ { x ^ { 2 } } & { \text { for } } & { x > 1 } \end{array} \right.$ Calculate $f(0)$ , $f(1)$ , and $f(3)$ .
$g ( x ) = \left\{ \begin{array} { c c c } { \sqrt { x } } & { \text { for } } & { x < 3 } \\ { 3 - x } & { \text { for } } & { x \geq 3 } \end{array} \right.$ Calculate $g(1)$ , $g(3)$ , and $g(9.4)$ .
$h ( x ) = \left\{ \begin{array} { c c c } { - x } & { \text { for } } & { x \leq 0 } \\ { \frac { 5 } { x } } & { \text { for } } & { 0 < x \leq 1 } \\ { 6 - 2 x } & { \text { for } } & { x > 1 } \end{array} \right.$ Calculate $h(–3)$ , $h(0)$ , $h(0.5)$ , and $h(4)$ .
Sketch a graph of $y = h(x)$ .

1-22.

In order to mail a letter in the United States, postage must be paid based on the weight of the letter. Although rates are tied to the number of ounces, the U.S. Post Office does not allow for payments of partial ounces.

A graph showing the postage rates for letters weighing fewer than $5$ ounces for the year 2015 is shown at right. This is another example of a step function Homework Help ✎

First quadrant step graph, x axis labeled weight, ounces, scaled from 0 to 5, split y axis with 5 tick marks, labeled as follows: first, 0.49, third, 0.93, fifth, 1.37, with 5 horizontal steps at each y value tick mark, first between x values 0 & 1, at y value, 0.49, with each additional step increasing 1 tick mark, between consecutive integer x values.

How much would you pay for a letter weighing $2.9$ ounces? For $3$ ounces? For $3.1$ ounces?
Write a piecewise-defined function that determines the postage rates for letters weighing between $0$ and $5$ ounces. Let $x$ represent the weight in ounces, and $y$ represent cost in dollars.

1-23.

A semi-circular flag is shown attached to a “pole” at right. Homework Help ✎

Horizontal segment, with semicircle, with it's diameter centered on the segment, distance across semi circle labeled 6.

Imagine rotating the flag about its pole and describe the resulting three-dimensional figure. Draw a picture of this figure on your paper. To help you visualize this, use the 1-23 HW eToo.

Calculate the volume of the rotated flag.

1-24.

Sketch a graph of the piecewise-defined function $g ( x ) = \left\{ \begin{array} { l l } { 2 ^ { x } } & { \text { for } x \leq 2 } \\ { 3 x - 2 } & { \text { for } x > 2 } \end{array} \right.$ . 1-24 HW eTool Homework Help ✎

State the domain and range of $g$ .
Is $g$ continuous at $x = 2$ ? Explain.
Is $g$ continuous for all values of $x$ ?

1-25.

The parabola $y = -(x - 3)^2 + 4$ is graphed at right. Four trapezoids of equal width are inscribed for $1 ≤ x ≤ 5$ . Homework Help ✎

Use the combined area of these trapezoids to approximate the area under the parabola for $1 ≤ x ≤ 5$ .
Is this area greater or less than the true area under the parabola? Explain.

Downward parabola, vertex at point, (3, comma 4), segments connecting the curve points in order from left to right: (1, comma 0), (2, comma 3), (3, comma 4), (4, comma 3), & (5, comma 0), with area below the segments & above the x axis shaded.

1-26.

What is the exact value of each of the following trig expressions? Homework Help ✎

$\sin\left(\frac{5\pi}{3}\right)$
$\tan\left(\frac{7\pi}{6}\right)$
$\sec\left(\frac{5\pi}{4}\right)$
$\csc\left(\pi\right)$

1-27.

Sketch the graph of $y =\frac { 1 } { x }$ . 1-27 HW eTool Homework Help ✎

Why does this graph have a vertical asymptote? What is the equation of that asymptote?
State the equation of the horizontal asymptote.
Alter the equation y = $\frac { 1 } { x }$ so that the vertical asymptote is $x = 1$ and the horizontal asymptote is $y = 3$ .

Math Notes

Polynomial Division with Remainder

The examples below show two methods for dividing. Both have remainders which are written as fractions. $\large \frac { x ^ { 4 } - 6 x ^ { 3 } + 18 x - 1 } { x - 2 }$

Using long division:

$\require{enclose} \begin{array}{rll} x^3-4x^2-8x+2\ \\[-3pt] x-2 \enclose{longdiv}{x^4-6x^3+0x^2+18x-1}\kern-.2ex \\[-3pt] \underline{x^4-2x^3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phantom{00}} && \\[-3pt] -4x^3+0x^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phantom{0} \\[-3pt] \underline{\phantom{0}-4x^3+8x^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phantom{0}} \\[-3pt] \phantom{0}-8x^2+18x\ \ \ \ \ \ \ \ \\[-3pt] \underline{\phantom{0}-8x^2+16x \ \ \ \ \ \ \phantom{0}} \\[-3pt] {\phantom{0}2x-1 \ \ \phantom{0}} \\[-3pt] \underline{\phantom{0}2x-4 \ \ \phantom{0}} \\[-3pt] {\phantom{0}3 \ \ \phantom{0}} \\[-3pt] \text{ remainder } \uparrow \ \ \ \end{array}$

Final answer: $x^3-4x^2-8x+2+\frac{3}{x-2}$

Using an area model:

2 by 5 generic rectangle

top edge first $x^3$

top edge second $-4x^2$

top edge third $-8x$

top edge fourth $+2$

top edge fifthRemainder

left edge top $x$

interior top first $x^4$

interior top second $-4x^3$

interior top third $-8x^2$

interior top fourth $+2x$

interior top fifth $3$

left edge bottom $-2$

interior bottom first $-2x^3$

interior bottom second $+8x^2$

interior bottom third $+16x$

interior bottom fourth $-4$

interior bottom fifth blank

$x^4$ Below generic rectangle

$-6x^3$

$+0x^2$

$+18x$

$-1$

Answer: $x^3-4x^2-8x+2+\frac{3}{x-2}$

1-28.

Use polynomial division to rewrite each of the following rational expressions. Homework Help ✎

$\frac { x ^ { 3 } + 2 x ^ { 2 } - 3 x + 4 } { x + 3 }$
$\frac { x ^ { 4 } - 5 x ^ { 2 } + 3 x - 3 } { x - 2 }$

Calculus Third Edition

1.2.1What if the function is in pieces?

COLLABORATIVE LEARNING EXPECTATIONS

Intuitive Notion of Continuity

Domain and Range Notation

Polynomial Division with Remainder