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MAT091 :: Lecture Note :: Week 06
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### Introduction to Functions

A function is a "relationship" that maps each input into one and only one output.

"Relationship" is quoted because functions can be defined in a variety of ways. The following was copied from the Wikipedia.

```   "The mathematical concept of a function expresses the
intuitive idea of deterministic dependence between two
quantities, one of which is viewed as primary (the independent
variable, argument of the function, or its 'input') and the
other as secondary (the value of the function, or 'output').
```

In the Wikipedia's function definition, the word deterministic implies that every input to a function always results in the same output.

##### Terminology

The valid inputs to a function represents the domain of the function.

The range of a function are all the possible outputs.

Inputs are independent of the outputs, but the outputs are dependent on the inputs.

Functions are typically given one letter names. In many instances, inputs are labeled (named) 'x' and outputs are labeled (named) 'y'.

```              +----------+
input ---> | function | ---> output
x        +----------+         y

y = f(x)   or    f(x) = y

y is the output, x is the input, f() is the function name
```

The following is a function with an implementation.

```   f(x) = 2x                   [each input is multiplied by 2]

f(-2) outputs -4            [input is -2]
f(-1) outputs -2            [input is -1]
f(0) outputs 0              [input is 0]
f(1) outputs 2              [input is 1]
f(2) outputs 4              [input is 2]

f() is a function because each input produces only one output.
Notice how each output value depends on the input value.

The domain of f() is all real numbers.
The range of f() is all real numbers.

Function f() could be named doubler() because the output
is always double (or 2 times) the input.

doubler(5) outputs 10
doubler(-10) outputs -20
doubler(5 * 3) outputs 30
doubler(1 - 2 - 3) outputs -8
```

Public domain dot-png from Wikipedia.org...

##### Are They Functions?
```   Is f() a potential function?  If yes, what's its domain and range?

+-------+
1 ----> |  f()  | ---->  -1          f(1) = -1            (1, -1)
+-------+
+-------+
2 ----> |  f()  | ---->  -2          f(2) = -2            (2, -2)
+-------+
+-------+
3 ----> |  f()  | ---->  -3          f(3) = -3            (3, -3)
+-------+

Is g() a potential function?  If yes, what's its domain and range?

+-------+
1 ----> |  g()  | ---->  -1          g(1) = -1            (1, -1)
+-------+
+-------+
2 ----> |  g()  | ---->  -2          g(2) = -2            (2, -2)
+-------+
+-------+
3 ----> |  g()  | ---->  -2          g(3) = -2            (3, -2)
+-------+

Is h() a potential function?  If yes, what's its domain and range?

+-------+
1 ----> |  h()  | ---->  -1          h(1) = -1            (1, -1)
+-------+
+-------+
2 ----> |  h()  | ---->  -2          h(2) = -2            (2, -2)
+-------+
+-------+
2 ----> |  h()  | ---->  -3          h(2) = -3            (2, -3)
+-------+
+-------+
3 ----> |  h()  | ---->  -4          h(3) = -4            (3, -4)
+-------+

Is Q() a potential function?  If yes, what's its domain and range?

+-------+
1 ----> |  Q()  | ---->  -1          Q(1) = -1            (1, -1)
+-------+
+-------+
2 ----> |  Q()  | ---->  -2          Q(2) = -2            (2, -2)
+-------+
+-------+
2 ----> |  Q()  | ---->  -2          Q(2) = -2            (2, -2)
+-------+
+-------+
3 ----> |  Q()  | ---->  -3          Q(3) = -3            (3, -3)
+-------+
```

If a function receives an input that is not in its domain, then the output of the function is "undefined." (i.e. A function does not work on inputs that are not in its domain.) If the domain of a function is not stated, then the its domain is all real numbers.

##### Related BABs and External Hyperlink(s)

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### Terminology Related To Functions

A function takes one input value and produces one output value. The same input value always produces the same output value.

```   input     domain    indepedent     horizontal-axis     x-axis
output    range     dependent      vertical-axis       y-axis
```

If a specific domain is not given for a function, then its domain is all real numbers.

If a function receives an input that is not in its domain, then the output of the function is undefined (i.e. the function doesn't work).

The output depends on the input. The input is independent of the output.

For a given domain, a function always produces the same range.

```   f(x) is read "f of x"     [not f times x]
g(n) is read "g of n"     [not g times n]
h(t) is read "h of t"     [not h times t]

f(x), g(n), h(t) all represent an output value

f(x) is a function of x   [f(x) depends on x]
g(n) is a function of n   [g(n) depends on n]
h(t) is a function of t   [h(t) depends on t]
```

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### Constant Function

A constant function is a function that produces the same output regardless of the input.

```   f(x) = k, where k is a constant
```

The graph of a constant function is a horizontal line.

If zero is in the function's domain, then the constant function intersects the vertical-axis at its constant output value.

```   f(x) = 3

f(-2) = 3
f(0) = 3
f(2) = 3

The graph for f() is a horizontal line that intersects
the vertical-axis at the point (0,3).

| 5
|
A------B------C        A: (-2,3)   B: (0,3)   C: (2,3)
|
|
-------+-------
-2     |      2
|
|
|
| -5
```

Unless stated otherwise, the domain of a constant function is all real numbers. The range of the constant function is the constant value that it always outputs.

##### Example of a Constant Functon

The following is the Speed At Stop Sign Function (SASSF).

```   f(s) = 0 MPH

's'  is the speed at which a vehicle approaches a stop sign
f(s) is the speed a vehicle should be moving at a stop sign

's' and f(s) are both in units of MPH (Miles Per Hour)

domain:  0 < s < infinity
range:  0

note:  For this specific constant function, the output does
not depend on the input.

note:  As of 8 June 2007, the land speed record is 763 MPH;
therefore, the practical domain for this function
could be 0 < s < 763.
```
##### Constant Functions are Linear Functions

A constant function is a linear function having zero slope.

```   f(x) = mx + b for a constant function is f(x) = 0x + b
```

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### Identity Function

A identity function is a function that returns its input for all valid inputs.

```   f(x) = x
```

The graph of an identity function is straight line having a slope of one.

If zero is in the function's domain, then the indentity function intersects the vertical-axis at point (0,0).

```   f(x) = x              [implementation]

f(-2) = -2
f(-1) = -1
f(0) = 0
f(1) = 1
f(2) = 2

The graph for f() is a straight line having a slope of
one that intersects the vertical-axis at the point (0,0).

| 5
|
|
|     E             E: (2, 2)
-2     |  D                D: (1, 1)
------C------             C: (0, 0)
B  |     2             B: (-1, -1)
A     |                   A: (-2, -2)
|
|
| -5
```

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### Absolute Value Function

The absolute value function outputs the absolute value of its input.

```   f(x) = |x|
```

The graph of an absolute value function, where the domain is all real numbers, is two staight lines that intersect at point (0,0). There is a straight line in the 2nd quadrant that has a slope of -1 and there is a straight line in the 1st quadrant having a slope of 1.

```   f(x) = |x|

f(-4) = 4
f(-2) = 2
f(0) = 0
f(2) = 2
f(4) = 4

| 5
A            |            E          A: (-4,4)    E: (4,4)
|
B      |      D                B: (-2,2)    D: (2,2)
|
-------------C-------------          C: (0,0)
-4            |            4
|
|
|
| -5
```

If the domain of an absolute value function is all real numbers, then its range is all real numbers greater than or equal to zero.

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### Ordered Pairs

The inputs and outputs of a function are often represented as ordered pairs (or points).

```   (input, output) ...or... (x, y) ...or... (x, f(x))
(t, n) ...or... (t, f(t))
(n, o) ...or... (n, f(n))
(a, b) ...or... (a, f(a))
```

The input is always the first value recorded followed by its respective output.

Data contained in tables can sometimes be represented as ordered-pairs.

```   input:  month ......... output:  #days

month |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | 10 | 11 | 12
------------------------------------------------------------------
#days | 31 | 28 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 | 31

(1, 31)   (2, 28)  (3, 31)  (4, 30)   (5, 31)  (6, 30)
(7, 31)   (8, 31)  (9, 30)  (10, 31)  (11,30)  (12, 31)

#days depends on the month (i.e. the output depends on the input)

The function  f(m)  takes a month as input and outputs
the number of days in that month.

f(m) = n

f(1) = 31
f(2) = 28
f(7) = 31
f(12) = 31
```

Ordered-pairs are points that can be graphed. The input is along the horizontal-axis and output is along the vertical-axis.

The following ordered-pairs could represent a function because all of the inputs are unique (i.e. different).

```   (-2, 3)  (0, 4)  (5, -2)  (7, 4)  (11, 7)
```

The following ordered-pairs could represent a function because all of the inputs are unique although some of the inputs repeat.

```   (-2, 3)  (0, 4)  (5, -2)  (7, 4)  (11, 7)  (0, 4)
```

The following ordered-pairs cannot be a function because input 3 produces different outputs (2 and 5).

```   (-2, 3)  (0, 4)  (-2, 3)  (3, 2)  (0, 4)  (3, 5)
```

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The graph formed by the intersection of a horizontal line and a vertical line can be divided into four quadrants.

```      vertical-axis (y-axis; outputs)
|
2 (II)   |     1 (I)
|
-----------o----------- horizontal-axis (x-axis; inputs)
|                  o is the origin (0, 0)
3 (III)  |     4 (IV)
|
```

The top-right quadrant is always the first quadrant with the remining quadrants numbered in a counter-clockwise direction.

Quadrants are sometimes numbered (labeled) using Roman Numerals.

Ordered-pairs (i.e. points or coordinates) are plotted (graphed) as follows.

```   (+x, +y) ... quadrant 1 (I)
(-x, +y) ... quadrant 2 (II)
(-x, -y) ... quadrant 3 (III)
(+x, -y) ... quadrant 4 (IV)
( 0,  0) ... origin
( 0, +y) ... vertical axis; above the origin
( 0, -y) ... vertical axis; below the origin
(+x,  0) ... horizontal axis; right of origin
(-x,  0) ... horizontal axis; left of origin
```

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### Using Tables To Represent Functions

When it comes to graphing functions, tables and other forms of data, input values are scaled along the horizontal-axis and output values are scaled along the vertical-axis.

When it comes to naming variables, the letter `x` is often used to represent the input variable and the letter `y` is used to represent the output variable; however, any letters can be used when it comes to naming input and output variables.

```                                 Table A
===============================================================
input: year (y)     | 2001 | 2002 | 2003 | 2004 | 2005 | 2007
---------------------------------------------------------------
output: # x-rays (x) |  55     43     81     24     43     61

The # of x-rays is a function of the year; i.e. year is the input
and # of x-rays is the output.  For this function, the input
variable is named 'y' and the output variable is named 'x'.

Table A expressed using function notation.

f(y) = x
f(2001) = 55
f(2002) = 43
f(2003) = 81
...
f(2000) = undefined  [cannot extrapolate]
f(2006) = undefined  [cannot interpolate]

Table B
==============================================================
input: hours (h)     | 1  |  2  |  3  |  4  |  5  |  6  |  7
--------------------------------------------------------------
output: minutes (m)   | 60 | 120 | 180 | 240 | 300 | 360 | 420

Number of minutes is a function of number of hours.
One hour is 60 minutes, two hours is 120 minutes, etc.

Table B expressed using function notation.

g(h) = m = 60(h)
g(1) = 60
g(4) = 240
g(7) = 420
g(2.5) = 150   [interpolation]
g(5.25) = 315  [interpolation]
g(0) = 0       [extrapolation]
g(8) = 480     [extrapolation]
```

`[definition]` A function is a "relation for which each element of the domain corresponds to exactly one element of the range."

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### Increasing and Decreasing Functions

Functions are sometimes categorized as either increasing or decreasing.

An increasing function is a function whose output values increase (i.e. get larger) when the inputs increase.

```   f(0) = 5
f(1) = 10
f(2) = 15
f(3) = 20

f(x) appears to be an increasing function
```

A decreasing function is a function whose output values decrease (i.e. get smaller) when the inputs increase.

```   g(0) = 100
g(1) = 90
g(2) = 80
g(3) = 70

g(x) appears to be a decreasing function
```

A constant function `f(x) = k` (where 'k' is a constant) is neither increasing nor decreasing.

The identity function `f(x) = x` is an increasing function.

Some functions can be both increasing and decreasing. For example, over an interval A, a function might be increasing, while it decreases over an interval B. Review the absolute value function.

```   abs(x) is decreasing from (-INF, 0)
abs(x) is zero at 0
abs(x) is increasing from (0, INF)
```

Technical definitions for increasing and decreasing functions.

```   increasing function:
====================
Function f(x) increases on an interval I if f(b) > f(a)
for all b > a, where a,b are in I.

decreasing function:
====================
Function f(x) decreases on an interval I if f(b) < f(a)
for all b > a, where a,b are in I.
```

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### Linear Functions

M-W.com defines linear as follows.

```   "of, relating to, resembling, or having a graph that
is a line and especially a straight line"

[...and...]

"having or being a response or output that is directly
proportional to the input"
```

Observe that the word linear contains the word line.

A linear function is a function is a "first degree polynomial function of one variable. These functions are called 'linear' because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line."

The following are equations for a line.

```   slope-intercept form:  y = mx + b
... used when slope and y-intercept are known

point-slope form:  y - y1 = m(x - x1)
... where (x1, y1) is a point on the line having slope m

standard form:  Ax + By = C
```

Slope is another way of saying "rate of change" and linear functions have a constant rate of change.

Slope is often described as the ratio "rise over run."

Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.

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### Slope (Linear Functions)

The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.

The following is how to find the slope between two unique points on a line.

```   point 1:  (x1, y1)
point 2:  (x2, y2)

slope = m = (y2 - y1) / (x2 - x1)

note:  x1 ≠ x2
```

Slope represents the ratio of change-in-output over change-in-input.

```   m = Δy / Δx    (change in output / change in input)

Δ is the "delta" character
Δy = y2 - y1
```

Slope is often described as the ratio "rise over run."

```
rise
m = ------
run
```

Some slope notes.

```   horizonal lines have slope 0   (y = f(x) = k, where k is a consant)
vertical lines have undefined slope   (x = k, where k is a consant)
parallel lines have equal slopes     (m1 = m2)
perpendicular lines have negative inverse slopes  (m2 = -1/m1)
```

The following is a verbose form the slope-intercept equation.

```   output = slope * input + initial_value_if_any

...or...

output = average_rate_of_change * input + initial_value_if_any
```

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### Intercepts (Linear Functions)

We frequently need to know the points where lines (graphs in general) intersect (cross) the horizonal-axis and vertical-axis (often called the x-intercept and y-intercept, respectively).

The vertical-intercept for a function is found by evaluating the function when its input is zero. The orderd-pair for a vertical-intercept will be `(0, something)`.

The horizontal-intercept for a function is found by setting its output to zero and finding what input value results in that output. The orderd-pair for a horizonal-intercept will be `(something, 0)`.

```   f(x) = 5(x) + 3

vertical-intercept (y-intercept)...
set the input to 0 and evaluate:  f(0) = 5(0) + 3 = 3
vertical-intercept is (0, 3)

horizontal-intercept (x-intercept)...
set the output to 0 and solve:  0 = 5(x) + 3
-3 = 5(x)
-3/5 = x
horizontal-intercept is (-3/5, 0)
```

Generalizations.

```   given  f(x) = m(x) + b

vertical-intercept:  (0, b)        [notice input is 0]
horizontal-intercept:  (-b/m, 0)     [notice output is 0]
```

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### Exercises: Finding Equations of Lines

Memorize the following.

```   slope formula:  m = (y2 - y1) / (x2 - x1)
slope-intercept form:   y = mx + b
point-slope form: y - y1 = m(x - x1)
```

`[exercise]` Find the equation of the line having a slope of `5` that passes through the point `(2, 8)`.

```   step 1:  substitute the slope 5 into  y = mx + b
y = 5(x) + b

step 2:  substitute 8 for y and 2 for x and solve for b
8 = 5(2) + b
8 = 10 + b
-10 + 8 = -10 + 10 + b
-2 = b

step 3:  rewrite the equation from step #1 and substitute -2 for b
y = 5(x) + -2
```

`[exercise]` Given the two points `(1, 5)` and `(3, 15)`, find the equation of the line.

```   step 1:  calculate the slope m
(15 - 5) / (3 - 1) = 10 / 2 = 5

step 2:  substitute one of the points and the slope into the
point-slope form equation for a line

y - 5 = 5(x - 1)       # point (1, 5) selected; m=5
y1=5, x1=1

step 3:  solve for y

y - 5 = 5x - 5         # distributive property on right-side
y     = 5x - 5 + 5     # add 5 to both sides
y     = 5x + 0         # this is slope-intercept form
```

Since the 'b' value is zero we know that this line passes through point `(0, 0)`. Recall, the 'b' value is the y-intercept (vertical-intercept) and it is obtained by setting the input to zero.

Solving this problem using the TI-83 calcuator.

```   press ON
press STAT
press 1:EDIT
if there are values in L1, then
left arrow to L1 column, if necessary
up arrow to L1 cell, if necessary
press CLEAR
press ENTER
if there are values in L2, then
up arrow to L1 cell
right arrow to L2 cell
press CLEAR
press ENTER
if necessary, use arrow keys to get to line 1 of L1
1 ENTER
3 ENTER
right arrow
5 ENTER
15 ENTER

There are now two point entered.

L1    |     L2
--------------
1     |     5
3     |     15

press STAT
right arrow to CALC
press 4:LinReg(ax + b)

home screen says:  LinReg(ax + b) [cursor]

press 2ND 1 (L1)
press COMMA (,)
press 2ND 2
press COMMA (,)
press VARS
right arrow to Y-VARS
press 1:Function
press 1:Y1

home screen says:  LinReg(ax + b) L1,L2,Y1 [cursor]
press ENTER

home screen says:  LinReg
y = ax+b
a = 5
b = 0
r^2 = 1
r = 1

press 2ND GRAPH to see a table
press GRAPH to see a graph of the line
press Y= to see the linear equation
```

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