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# Section 3.17Properties of Functions

## Basic Terminology

The table below summarizes basic terminology developed so far in the book. It also introduces new terms and meanings regarding properties of functions. The new meanings will be discussed more fully following the table.

Term Symbol Meaning
Interval of numbers from a to b $\,[a,b]$ if you include a and b
$\,(a,b)$ if you exclude a and b
All real numbers between a and b. The interval is called closed if you include a and b; the interval is called open if you exclude a and b.

Intervals can be half open or half closed, denoted $[a,b)$ when you include a and exclude b, or $(a,b]$ when you exclude a and include b.
Variable Conventionally, any letter at the upper end of the alphabet, such as x, y, or z But any notation can be used as a variable as long as you so state it. If you represent the value of a quantity whose measure varies within a situation, then you are using that letter as a variable.
Independent variable The variable whose values you control, or the variable whose values you take as determining the other variable's values.

Dependent variable

The variable whose values are determined by values of the independent variable.
Function Any letter, symbol, or word declared as representing a function A relationship between an independent variable and a dependent variable is called a function if each value of the independent variable is related to exactly one value of the dependent variable.

Functions can be defined by a rule for computing its values, such as $f(x)=3x^2-\cos(x)$ or conceptually, such as "Let $\text{dist}(x)$ be the number of meters from start run by Usain Bolt x seconds after the start in the final heat of the 2012 Summer Olympics."

In both cases, every value of x is associated with exactly one value $f(x)$. The rule $x^2-\cos(x)$ always produces just one number for any value of x. Usain Bolt is never two distances from start at a given moment in time during the race.

Domain of a function All values of the function's independent variable for which the function has a value. $f(x)=1/x$ is not defined when $x=0$, so 0 is not in the domain of f.

Range of a function

All values related by the function to values of the independent variable.
One-to-one function 1-1 A function f is said to be one-to-one when no two values of its independent variable are related to the same value of its dependent variable. In symbols, if a and b are in the domain of f and $a\ne b$, then $f(a)\ne f(b)$.
Discontinuous function A function f is said to be discontinuous at a value $x_0$ in its domain if there is a tolerance level $\epsilon > 0$ such that there is no interval around $x_0$ having all values of $f(x)$ within that tolerance.

Put another way, f is discontinuous at a value $x_0$ in its domain if there is a value of $\epsilon$ greater than 0 so that in any interval containing $x_0$, there are values $x_1$ and $x_2$ in the interval such that $|f(x_1)-f(x_2)|\gt \epsilon$.
Continuous function A function is said to be continous over an interval if it is not discontinuous at any value in the interval.
Increasing function A function f is said to be increasing over an interval if $f(a)\lt f(b)$ whenever $a\lt b$ for all values a and b in the interval.
Decreasing function A function f is said to be decreasing over an interval if $f(a)\gt f(b)$ whenever $a\lt b$ for all values a and b in the interval.

Even function

A function f is said to be even if $f(-x)=f(x)$ for all values of x in its domain. The graph of an even function in the Cartesian coordinate system is symmetric by a reflection about the y-axis.

Odd function

A function f is said to be odd if $f(-x)=-f(x)$ for all values of x in its domain. The graph of an odd function in the Cartesian coordinate system is symmetric by a rotation of 180° about the origin.

## Examples and Discussion

### Function

Do not equate "function" with "rule"!

A function definition might include an explicit rule of assignment, but it needn't. A function can also be defined conceptually, as in the example already given: "Let $\text{dist}(x)$ be the number of meters from start run by Usain Bolt x seconds after the start in the final heat of the 2012 Summer Olympics." In this race, Usain Bolt was exactly one distance from start at each moment during the race. We can say this even though we do not have an explicit rule to determine what number of meters he was from start at any moment during the race. The expression $\text{dist}(3.27)$ has a definite meaning and a definite value even though, at this moment, we have no way to determine the value.

There is a way to say "Any value of x is associated with exactly one value of y" formally:

Suppose you are told there is a relationship between values of x and y named g and that g is a function from x to y. If you are later told $g(a)\ne g(b)$, then you can conclude automatically that $a\ne b$.

How does this capture "... exactly one ..."? Because if $g(a)\ne g(b)$ while $a=b$, then this says $g(a)\ne g(a)$ (since $a=b$). One value of x would be associated with two values of y.

Take $g(x)=\pm \sqrt x$ for $x\ge 0$. We have, for example, $g(4)=-2$ and $g(4)=2$. By definition, g is not a function from x to y even though it relates values of x and y. There is at least one value of x (namely, 4) that is associated with two values of y, and therefore $x=4$ is not associated by g with exactly one value of y.

### One-to-one Function

Here is another way to say a function f is 1-1: f is 1-1 if every value in the range of f is associated with exactly one value in the domain of f.

### Discontinuous Functions

The idea of a function f being discontinuous at $x=a$ is that there is a "gap" in values of $f(x)$ for values of x around $x=a$.

Put another way, there is some level of tolerance for which you cannot approximate a value of $f(a)$ with the assurance that every value of $f(x)$ will be within this tolerance for all values of x in any interval containing $a$.

A simple example is $$f(x)= \left\{ \begin{array}{ll} x & \text{if } x \leq 1 \\[1ex] x+0.0000001 & \text{if } x \gt 1 \end{array} \right.$$ If we set our tolerance at 0.00000005 and let $x_0=1$, then no interval around $x_0=1$ will have all values of x in it having $f(x)$ within this tolerance of $f(1)$.

Figure 3.17.1 illustrates how graphs can be deceiving. The graph of $y=f(x)$ (f defined as above) appears to be continuous around $x=1$. The graph shows a vertical line at $x=1$ and two horizontal lines, one at $y=1.0000005$ and one at $y=0.99999995$. These blue lines show a tolerance of 0.0000005 around $f(1)=1$.

Figure 3.17.1. A function whose graph at one scale suggests it is continuous everywhere turns out to be discontinuous at $x=1$. No interval around $x=1$ has all values $f(x)$ within 0.0000005 of $f(1)$ for values of x in the interval.

### Continuous Functions

A function f is continuous over an interval if there is no value $x_0$ of x in the interval at which $f(x_0)$ is discontinuous. Put loosely, there are no "gaps" in the values of $f(x)$ over the interval.

Notice that we speak of discontinuity at a value of x, but we speak of continuity over an interval of values of x. The idea of discontinuity indeed involves intervals, but discontinuity happens at specific values in an interval.

If a function is continuous over an interval, we say it is continuous at each value in the interval.

### Increasing Functions

A more colloquial statement is, "A function is increasing over an interval if the value of the function increases as the value of its argument increases."

An equivalent, but less common, way to say a function increases over an interval is that the value of the function decreases as the value of its argument decreases.

### Decreasing Functions

A more colloquial statement is, "A function is decreasing over an interval if the value of the function decreases as the value of its argument increases."

An equivalent, but less common, way to say a function decreases over an interval is that the value of the function increases as the value of its argument decreases.

## Activities

In these activities you will see a video showing values of x related to values of y by a function named f. Each video is followed by questions for you to answer.

A note about the logic of explaining an answer of "No" in these activities: You must support a "No" answer to a question about a property by giving one specific example where the property is not satisfied.

For example, to justify "no" as your answer to the question, "Is f one-to-one over its domain?" you must state two specific values of x, say 2 and 4, and state that $f(2)=f(4)$ while $2\ne 4$. A property fails to hold for a relationship when it fails once.

### Activity 3.17.1

The animation below shows values of x being related to values of y by a relationship named f. Play the video, then answer the questions following the video using meanings for terms in the question. Click here to remind yourself of terms' meanings. You can replay the video and pause it, or scroll the play bar to examine parts of the video. (Move your pointer away from the video to remove the scroll bar.)

1. What, approximately, is the domain of f?
2. What, approximately, is the range of f?
3. Is the relationship named f a function from x to y? Explain.
4. Is f continuous over its domain? Explain.
5. Is f one-to-one over its domain? Explain.
6. Over what interval(s) in its domain, if any, is f increasing? Explain.
7. Over what interval(s) in its domain, if any, is f decreasing? Explain.

### Activity 3.17.2

The animation below shows values of x being related to values of y by a relationship named f. Play the video, then answer the questions following the video using meanings for terms in the question. Click here to remind yourself of terms' meanings. You can replay the video and pause it, or scroll the play bar to examine parts of the video. (Move your pointer away from the video to remove the scroll bar.)

1. What, approximately, is the domain of f?
2. What, approximately, is the range of f?
3. Is the relationship named f a function from x to y? Explain.
4. Is f continuous over its domain? Explain.
5. Is f one-to-one over its domain? Explain.
6. Over what interval(s) in its domain, if any, is f increasing? Explain.
7. Over what interval(s) in its domain, if any, is f decreasing? Explain.

### Activity 3.17.3

The animation below shows values of x being related to values of y by a relationship named f. Play the video, then answer the questions following the video using meanings for terms in the question. Click here to remind yourself of terms' meanings. You can replay the video and pause it, or scroll the play bar to examine parts of the video. (Move your pointer away from the video to remove the scroll bar.)

1. What, approximately, is the domain of f?
2. What, approximately, is the range of f?
3. Is the relationship named f a function from x to y? Explain.
4. Is f continuous over its domain? Explain.
5. Is f one-to-one over its domain? Explain.
6. Over what interval(s) in its domain, if any, is f increasing? Explain.
7. Over what interval(s) in its domain, if any, is f decreasing? Explain.

### Activity 3.17.4

The animation below shows values of x being related to values of y by a relationship named f. Play the video, then answer the questions following the video using meanings for terms in the question. Click here to remind yourself of terms' meanings. You can replay the video and pause it, or scroll the play bar to examine parts of the video. (Move your pointer away from the video to remove the scroll bar.)

1. What, approximately, is the domain of f?
2. What, approximately, is the range of f?
3. Is the relationship named f a function from x to y? Explain.
4. Is f continuous over its domain? Explain.
5. Is f one-to-one over its domain? Explain.
6. Over what interval(s) in its domain, if any, is f increasing? Explain.
7. Over what interval(s) in its domain, if any, is f decreasing? Explain.

## Exercise Set 3.17

1. Sketching graphs in the Cartesian coordinate system:
1. Sketch a graph of the relationship between values of x and values of y for the video in Activity 1.
2. Sketch a graph of the relationship between values of x and values of y for the video in Activity 2.
3. Sketch a graph of the relationship between values of x and values of y for the video in Activity 3.
4. Sketch a graph of the relationship between values of x and values of y for the video in Activity 4.
2. The animation below simulates a car driving on a race course. The car traverses the course 4 times. Quantities mentioned in a-c are in the order dependent quantity followed by independent quantity.
1. Consider the car's number of miles driven from start in relation to number of minutes elapsed since starting. Is this relationship a function? Is it continuous over its domain? Is it 1-1 over its domain?
2. Consider the car's position on the track in relation to number of minutes elapsed since starting. Is this relationship a function? Is it continuous over its domain? Is it 1-1 over its domain?
3. Consider the car's straight-line distance from start in relation to distance driven from start. Is this relationship a function? Is it continuous over its domain? Is it 1-1 over its domain?
3. You are told the graph of $y=h(x)$ is symmetric about the line $x=4$.
1. Define a function j so that $h(j(x))$ is even. Try your general solution with $h(x)=x^4-16x^3+96x^2-256x+260$. (Click here to get a GC file already set up for you.
2. Suppose the graph of $y=k(x)$ is symmetric about the line $x=c$. Define a function j so that $k(j(x))$ is even. Explain your answer.
4. Play the video shown below or scroll through it by moving the play bar.

1. What, approximately, is the domain of f?
2. What, approximately, is the range of f?
5. Let h be defined as $h(x)=1/(2-x)$.
1. What is the domain of h? The range of h?
2. Is h continuous over its domain? Remember, for h to be discontinuous at a value $x_0$ in its domain there must be a gap in y-values over an interval containing $x_0$ and the interval must be contained in h's domain.
6. State whether each function is even, odd, or neither even nor odd. Explain your answers. Graph $y=\_\_(x)$ and $y=\_\_(-x)$ [insert the function name] in GC if you are unsure of your answer.
1. $f(x)=\sin(x)$
2. $g(x)=\cos(x)$
3. $h(x)=\tan(x)$ Recall: $\tan(x)=\dfrac{\sin(x)}{\cos(x)},\, \cos(x)\ne 0$
4. $j(x)=x^2-4x+4$
5. $k(x)=x\sin(x)$
6. $m(x)=x\cos(x)$
7. $n(x)=3x+1$
7. The function f is even. The function g is odd. Both are defined for all real numbers. What can you conclude about:
1. $h(x)=f(x)+g(x)$
2. $k(x)=f(x)g(x)$
3. $l(x)=f(g(x))$
4. $m(x)=g(f(x))$

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