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MAT091 :: Lecture Note :: Week 02
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### Like Terms

Evaluating expressions and solving equations can sometimes be made "easier" if they are "simplified." One step in simplifying expressions/equations is the "combining" of like-terms. Like-terms are terms having the same variables raised to the same exponents.

```   5x + 2x = 7x
2x - 5x = 2x + (-5x) = -3x
x2 + 4x2 = 5x2 [x2 = 1x2]
9x10 + 9y10 are not like-terms [different variables]
5x2 and 2x are not like-terms [different exponents]
```

wyzant.com:: Simplifying Expressions Calculator

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### Algebraic Expressions

An algebraic expression is a mathematical expresssion that contains at least one variable along with zero or more numbers (i.e. constants) and zero or more arithmetic operators.

The following are algebraic expressions.

```   n
3n
5n + 10
2x^2 - 2y^3
9a + 2b - 3c
11(x + y^-3 + 2w)
```

Variables are unknown values that are represented by single-character letters. For example, `3n` is `3 times 'n'`, where 'n' is a variable. If a value is assigned to 'n', then the expression can be evaluated.

```   3n + 8

if n = 3, then 3(3) + 8 = 100
if n = -5, then 3(-5) + 8 = -7
if n = 0, then 3(0) + 8 = 8
```

The replacement (or swapping) of variables with values can be described in a variety of ways: Values can be "plugged into" or "assigned to" variables; or variables can be "replaced with" or "substituted with" values; or we "let" variables be certain values.

```   5x + 1  assign 5 to x ........... 5(5) + 1 = 25 + 1 = 26
5x + 1  replace x with 5 ........ 5(5) + 1 = 25 + 1 = 26
5x + 1  plug 5 into x ........... 5(5) + 1 = 25 + 1 = 26
5x + 1  substitute x with 5 ..... 5(5) + 1 = 25 + 1 = 26
5x + 1  let x be 5 .............. 5(5) + 1 = 25 + 1 = 26
```

Again, expressions are typically evaluated after variables have been assigned (i.e. given) values.

If a variable is multiplied by a constant, then the constant is called a coefficient.

```   3n ...... 'n' is a variable and 3 is a coefficient
x ....... 'x' is a variable and 1 is the coefficient
-1y ..... 'y' is a variable and -1 is the coefficient
```

The combination of a coefficent and variable is called a factor. Factors are numbers, variables, and expressions that are multiplied together to produce a product.

Factors separated by addition and subtraction operators are called terms.

```   3n + 1 .............. 1 factor, 2 terms
4a + 2b - 10 ........ 2 factors (4a and 2b), 3 terms
7x - 5 + 8y - 11 .... 2 factors (7x and 8y), 4 terms
2x^3 + 10^2.......... 1 factor (2x^3), 2 terms
3(4) - 2(1 + 3)...... 2 factors, 2 terms
```

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### Simplifying Expressions

```   a.  Simplify:  4 - 2(x + 3)
```
```       4 + -2(x + 3)     # rewrite a - b as a + -b
4 + -2x + -6      # distribute -2
-2x - 2           # combine like terms (the constants)
-2x + -2          # rewrite as an addition problem
-2(x + 1)         # factor out a -2

check work:  4 - 2(x + 3)               -2(x + 1)

let x = 0... 4 - 2(0 + 3)               -2(0 + 1)
4 - 2(3) = 4 - 6 = -2      -2(1) = -2

let x = 1... 4 - 2(1 + 3)               -2(1 + 1)
4 - 2(4) = 4 - 8 = -4      -2(2) = -4

let x = -1... 4 - 2(-1 + 3)             -2(-1 + 1)
4 - 2(2) = 4 - 4 = 0      -2(0) = 0

check work using the calculator:
press Y=
enter 4-2(x+3) into y1
enter -2(x+1) into y2
press GRAPH to see one line
press 2nd GRAPH to see table values
press Y=
change y2 to -2(x+2)
press GRAPH to see two lines
press 2nd GRAPH and compare table values

```
```   b.  Simplify:  (x^2 + x - 5) - (2x^2 - x + 3)
```
```       # convert subtractions into additions
(x^2 + x + -5) + (-1)(2x^2 + -x + 3)

x^2 + x + -5 + -2x^2 + x + -3      # drop ()s and distribute -1
x^2 + -2x^2 + x + x + -5 + -3      # combine like terms (step 1)
-x^2 + 2x + -8                     # combine like terms (step 2)

check work:  let x = 0
(x^2 + x - 5) - (2x^2 - x + 3)
(0^2 + 0 - 5) - (2(0^2) - 0 + 3) = -5 - 3 = -8

-x^2 + 2x + -8
-0^2 + 2(0) + -8 = 0 + 0 + -8 = -8

check work:  let x = 1
(x^2 + x - 5) - (2x^2 - x + 3)
(1^2 + 1 - 5) - (2(1^2) - 1 + 3)
(     -3    ) - (      4       ) = -7

-x^2 + 2x + -8
-1^2 + 2(1) + -8
-1  + 2    + -8 = -7

check work:  let x = -1
(x^2 + x - 5) - (2x^2 - x + 3)
((-1)^2 + -1 - 5) - (2(-1^2) - -1 + 3)
(   1   + -1 - 5) - (2       - -1 + 3)
(       -5      ) - (     6          ) = -11

-x^2 + 2x + -8
-(-1)^2 + 2(-1) + -8 = -1 + -2 + -8 = -11

check work using the calculator:
press Y=
enter (x^2 + x - 5) - (2x^2 - x + 3) into y1
enter -x^2 + 2x + -8  into y2
on the y2 line... left arrow to the \ and press ENTER
press GRAPH to see only one parabola
press 2nd GRAPH to compare table values

```

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### Define: horizontal and vertical

The horizon is the "apparent junction of earth and sky."

The following is a number line written horizontally.

```   <--+---+---+---+---+---+---+---+---+---+---+-->
-5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5
```

The vertical is "perpendicular to the plane of the horizon or to a primary axis." [Perpendicular is "being at right angles to a given line or plane."]

The following is a number line written vertically.

```      ^
|
+5 +
|
+4 +
|
+3 +
|
+2 +
|
+1 +
|
0 +
|
-1 +
|
-2 +
|
-3 +
|
-4 +
|
-5 +
|
v
```
##### Right Angle

Perpendicular lines intersect and their point of intersection is called the origin. Perpendicular lines form four right angles at the origin. A right angle measures ninety degrees (`90°`).

With respect the Cartesian coordinate system, the horizonal line is the `x-axis` and the vertical line is the `y-axis`.

```           | <-- vertical-axis (y-axis)
|
|
|
--------o-------- <-- horizonal-axis (x-axis)
|\
| \
|  o is the origin
|
```

Note: The Cartesian coordinate system was developed by French mathematician and philosopher Rene Descartes in approximately 1637.

RoadHacker:: Horizon/Vertical Slideshow [opens new window]

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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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### 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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