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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 16: FUNCTIONS

 

SECTION 16.1: 5 WAYS TO CONCEPTUALIZE A FUNCTION

In an earlier section, you were introduced to a very basic definition of a function. I stated that a function is a set of ordered pairs such that any given x-value in the set of ordered pair has only one unique y-value assigned to it. Below are 5 different ways to model the concept of a function.


Figure

Figure


Figure

SECTION 16.2: FUNCTIONS AND RESEARCH

Researchers often collect data as ordered pairs. Functions tend to be the "essence" of "real world" problems. What does this previous statement mean?

Researchers tend to explore situations where "function" relationships occur. For example, at a given instant in time, you have one weight. I'm sure at the time you wake up in the morning, say 6 a.m., you weigh a fixed amount, say 165 pounds. I don't think you could ever imagine an instance where you might weigh 165 pounds and 184 pounds at the same time 6 a.m. on the same morning.

How about this? What if someone said to you: "At 11:30 a.m. today, Mr. Landry will be in two different places at Deering High School." I don't think you would believe it, because you believe in the "functional nature" of our everyday experience. At a particular time t you know that I will be at a particular position P: that's our everyday experience that we depend on. Such accepted human perceptions on the "nature of the world" are the "essence" of the mathematical definition of a function.

Below is a table of ordered pairs collected by census collectors in Mexico over a seven-year period beginning in 1980 and a graph of these ordered pairs to the right of the table.


Figure

Of course the above graph of the research data is a function. But is there a quick way to determine a graph is a function without looking through all the ordered pairs? \

SECTION 16.3: VERTICAL LINE TEST TO DETERMINE IF A GRAPH IS A FUNCTION

If you have a graph of ordered pairs and you want to have a quick test to see if the graph is a function, place a vertical line over the graph. If the vertical line touches only one point of the graph no matter where you move that vertical line then you have a function. Consider the graphs of the same function below:

Figure

As you can see, as I move the vertical line from left to right it will only touch one point at any time, which implies that the t value does not have two different p values. This test for determining if a graph is a function is called the "VERTICAL LINE TEST."

SECTION 16.4: GRAPHS OF ORDERED PAIRS THAT ARE NOT FUNCTIONS

Examine the graph below. The set of ordered pairs on the graph below  are { (1,0), (0, 1), (-1, 0), (0, -1) }.

Figure

If you use the vertical line test on the graph above, you will see that a vertical line will go through two points when placed on the graph. Look below.

Figure

This vertical line above goes through two points (0, 1) and (0, -1). Can you see that we have a situation where an x-value, in this instance, x = 0, has two different y-values, y = 1 and y = -1. So we know that this graph is not a function from the Vertical Line Test. But we already knew this because we had the set of ordered pairs { (1,0), (0, 1), (-1, 0), (0, -1) } that showed us that an x-value had two different y-values.

SECTION 16.5: INVERSE OF A FUNCTION AND THE HORIZONTAL LINE TEST

If you have a function and swap the x and y values of each ordered pair of the function, and the resulting set of ordered pairs is a function, that resulting set of ordered pairs is called the "inverse" of the function.

Below is the table of values from a previous problem. The ordered pairs are { (0, 7), (1, 10), (2, 13),...,(5, 22) } If you swapped the t and d values you would get the set of ordered pairs { (7, 0), (10, 1), (13, 2),...,(22, 5) }. This resulting set of ordered pairs is a function so this resulting set of ordered pairs would be called the "inverse" of the function { (7, 0), (10, 1), (13, 2),...,(22, 5) }.

Here is another example of a function: MATH. Notice that each x-value has one and only one y-value. Now let's swap the ordered pairs in the function MATH. We would get: MATH. Notice that the resulting ordered pairs is NOT a function because the x-value 5 has two different y-values, 2 and 7. Therefore, we would say that the function MATH does not have an inverse.

IMPORTANT
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A function DOES NOT necessarily have an inverse.
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Examples: Which of the functions have an inverse?

a) MATH

Swap the ordered pairs: MATH - This is a function.

Therefore MATH has an inverse and it is MATH.

b) MATH

Swap the ordered pairs: MATH - This is not a function.

Since MATH and MATH are two x-values that are the same and have different y-values we say MATH does not have an inverse.

 

HORIZONTAL LINE TEST FOR DETERMINING
IF A FUNCTION HAS AN INVERSE

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If you are given a function as a graph then  there is a quick way to determine if it has an inverse is to use the "HORIZONTAL LINE TEST." 

HORIZONTAL LINE TEST
"if a horizontal line goes through more than one point on a function then the function does not have an inverse.

Below is the graph of $y=x^{2}$.

Figure

Notice that the horizontal line y = 2 goes through the two points  (-1,1) and (1,1) which are on $y=x^{2}$. Therefore y = x$^{2}$ does not have an inverse since (1, 1) and (-1, 1) would become (1, 1) and (1, -1) on the "inverse" which would produce two x-values that are the same but would have different y-values.

REMEMBER AN INVERSE MUST BE A FUNCTION

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SECTION 16.6: FUNCTION NOTATION

Some equations produce ordered pairs that are functions. If you produced ordered pairs from y = 4x - 3, you would get a set of ordered pairs that would fall on a straight line tilted to the right. Why? Because this is an equation in the form of y = mx + b which produces ordered pairs that follow the path of a line. And since the slope, m = 4, is positive the line tilts to the right.

If an equation is known to be a function, which y = 4x - 3 is, mathematicians use an alternative symbolism for the variable y or any other variable you so happen to use on the left-hand side of the equation that is a function. Compare the following two equations:


Figure

Notice the circled algebraic expressions above? Notice that each of those circled expressions are the same? Notice the symbols on the left hand side of these two equations are different? One equation contains the familiar y, while the other equation contains the lower case letter "f" followed by x in parentheses. Compare the following equations again:


Figure

Notice that the circled mathematical expressions above are the same? Notice that the symbols on the left-hand side of the equations are different? One equation contains the familiar y, while the second equation contains the lower case letter "f" followed by x in parentheses; the third, contains the lower case letter "w" followed by x in parentheses; the fourth, contains the lower case letter "h" followed by x in parentheses.

The symbols f(x), w(x), and h(x), above, all start with a lower case letter of the alphabet followed by parentheses that contain the same variable that is on the right hand side of each of the above equations. These symbols are examples of "function notation." If you see this type of symbol on the left hand side of an equation it means that the expression on the right hand side of the equation will produce ordered pairs that will in fact be a function.

So, if I gave you the following equation, $s(x)=2x^{2}+4x-5$, then I am letting you know that the algebraic expression $2x^{2}+4x-5$ will produce a set of ordered pairs that will be a function.

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"Function Notation" is used to let you know that the algebraic expression it is
assigned to will produce a set of ordered pairs that is a function.
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SECTION 16.7: INCORRECT USE OF FUNCTION NOTATION

In $s(t)=2t^{2}+4t-5$ the mathematical expression on the right-hand side contains the variable t and so does the function notation s(t). Notice that there is a t in the parentheses of s(t). I said that this must occur if you use function notation correctly. You would never see expressions like $s(r)=2m^{2}+4m-5$ or $g(p)=5x^{2}-7x+5$ in mathematics.


Figure

SECTION 16.8: FUNCTION NOTATION AS A DELIVERER OF INFORMATION FOR PRODUCING ORDERED PAIRS

Consider the following function $s(t)=2t^{2}+4t-5$ and the following notation MATH and $s(-5).$ Whether you realize it or not, you have 5 different sets of instructions!

Let's deal with the symbol $s(0)$ first.

The symbol "$s(0)$" is a set of instructions. Here are the instructions:

  • First, you are being asked to take the number inside of the parentheses of s(0), in this case 0, and assign it to t, the variable in the algebraic expression $2t^{2}+4t-5$. That is, let t = 0.

  • Second, you are being asked to take that number assigned to t, i.e. 0, and then substitute it into the algebraic expression that s(t) has been set equal to, in this case $2t^{2}+4t-5.$

That is, $2(0)^{2}+4(0)-5.$

  • Third, evaluate the algebraic expression, i.e. MATH.

  • Fourth, you assign the function notaion to the result of your evaluation, i.e. $s(0)=-5$.

  • Fifth, create an ordered pair using the value of t, i.e. 0, and the resulting evaluation, i.e. -5. That is, (0, -5)

 

You will use the same principle above to perform the following instructions: s(1), s(-4), s(7), and s(-5).

MATH Then your ordered pair will be (1, 1).

MATH Then your ordered pair will be (-4, 11).

MATH Then your ordered pair will be (7, 121).

MATH Then your ordered pair will be (-5, 25).

Interesting use of symbolism? On the next page let's compare and contrast x-y symbolism with function notation symbolism.

SECTION 16.9: COMPARING AND CONTRASTING "y =" SYMBOLISM WITH FUNCTION NOTATION SYMBOLISM

In the past you have seen equations like $y=2t^{2}+4t-5$ but not like $s(t)=2t^{2}+4t-5$. The only difference between these two equations is that one uses y and the other s(t). To create ordered pairs with $y=2t^{2}+4t-5$ you would choose an arbitrary number for t, substitute that number into the mathematical expression $2t^{2}+4t-5$, and then assign the result of your evaluation to y. See below:

Using y = notation: $y=2t^{2}+4t-5$: $t=3$.

Substitute t = 3 into $2t^{2}+4t-5$ to get MATH.

So when t = 3 the result for y is 25. We assign y = 25..

(3, 25) is the resulting ordered pair.

Using function notation: $s(t)=2t^{2}+4t-5.$ Find s(3).

t = 3: substitute 3 into $2t^{2}+4t-5$ to get MATH

So when t = 3 the result for $2t^{2}+4t-5$ is 25. We assign $s(3)=25.$

(3, 25) is the resulting ordered pair.

 

SECTION 16.10: INDEPENDENT AND DEPENDENT VARIABLES IN SCIENCE

In order for you to stay alive, you are "dependent" on air, water and food. Without these three things you cannot live. Your existence is "dependent" on some other things.

As you know the word "variable" means change. Your existence is variable because you change throughout your life.

We are dependent and we are variable: we are in fact a "dependent variable". As a "dependent variable" we are dependent on other things like food, water, and air, and these things change us through time.

The things that we depend on for our existence could be called "independent variables." See the chart below:

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AIR ---------> INDEPENDENT VARIABLE

YOU---------> DEPENDENT VARIABLE

 

WATER ---------> INDEPENDENT VARIABLE

YOU ---------> DEPENDENT VARIABLE

 

FOOD ---------> INDEPENDENT VARIABLE

YOU ---------> DEPENDENT VARIABLE

 

HYDROGEN --------> INDEPENDENT VARIABLE

WATER(H2O) --------> DEPENDENT VARIABLE

 

OXYGEN --------> INDEPENDENT VARIABLE

WATER(H2O) --------> DEPENDENT VARIABLE

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Researchers in the sciences are always searching out relationships between independent and dependent variables. Why? Because that understanding is the basis for us learning how to control our environment and how our environment controls us.

SECTION 16.11: INDEPENDENT AND DEPENDENT VARIABLES IN MATHEMATICS

When you have an equation like y = 3x + 5 or any equation for that matter, the variables x and y have special names assigned to them. Before you can get a y-value you need to have an x-value. In other words, the result assigned to y is dependent on the value assigned to x. You cannot get a y-value without an x-value.

Given what we know about independent and dependent variables in science, it does not seem unreasonable to say that x is the independent variable and y is the dependent variable in the equation y = 3x + 5.

You can't get a value for y without first being given a value for x. The value of y is in fact dependent on the value assigned to x.

If we used function notation like f(x) = 3x + 5 we would say x is the independent variable and f(x) is the dependent variable.

Examples) Name the independent and dependent variables in:

a) $s=4x+7$   b) $u(f)=3f^{3}+4f-2$
     
c) MATH   d) $d(e)=-3e^{2}+6e-12$

 

 

independent variable

 

 

dependent variable
a) x

 

 

s
b) f

 

 

u(f)
c) d

 

 

g
d) e

 

 

d(e)

SECTION 16.12: HOW TO SPEAK FUNCTION NOTATION

Now it seems reasonable after doing all this work with functions that you should learn how to pronounce something like f(x), g(x), q(x), or any function notation for that matter.

Function Notation You Say
a) f(x) f of x
b) g(x) g of x
c) r(x) r of x
d) m(x) m of x

SECTION 16.13: "IN TERMS OF..."

Another phrase you'll have to get use to when dealing with function notation is the phrase "in terms of." Let me give a few examples of this new language, and then we will pick that language apart.

ex. $g(x)=5x+7$ We would say that the function g is in terms of x.


Figure

ex. $m(u)=5u+7$ We would say that the function m is in terms of u.


Figure

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES