MAT092::Lecture Note::Week 04

Home	Previous	Next

Overview

[exercise] f is a function... write f(x) as an ordered-pair.
f(x) -- What Does It Mean?
Algebraic expressions.
Using [], [), (], () to specify intervals.
Review domain and range.
Introduction to increasing and decreasing functions.

Assessments:

Week #4 final exam review (printed copy only)
Fourth Assessment on Functions
Second Assessment on Exponents
First Assessment on Specifying Intervals

Quote of the Week {Furman.edu:: Mathematical Quotations Server}
CQ + PQ > IQ (curiosity, passion, intelligence quotients)
-- Thomas Friedman (01953-) {American journalist; "The World is Flat; more...} [future] [log]
BARS of the Week {NASA.gov:: Astronomy Picture of the Day}
Prescott AZ | Tempe AZ | Tucson AZ [log]

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

{TopOfPage} {101Science} {quiet} { TI-83} { Factors} {MathWords}

Solve 'x' When There's No Real Number Solution

There are times when an equation has no real number solutions.
3x = 3x + 1 subtract 3x from both sides: 3x - 3x = 3x - 3x + 1 0 = 1 0 = 1 is a false statement (i.e. it's not true!)

The Mathematics In Action book calls these situations contradictions (see section 2.17).

{TopOfPage} {101Science} {quiet} { TI-83} { Factors} {MathWords}

Using [], [), (], () To Specify Intervals

An interval is a set containing every number that is between given minimum and maximum values (end-points).

An interval of values can be specified using square brackets and parenthesis. The square bracket implies inclusion, while the parenthesis implies exclusion.

expressing an interval of values where m is the minimum value and n is the maximum value [m, n] implies the interval m through n, where both m and n are in the interval [m, n) implies the interval m through n, where m is in the interval, but n is not (m, n] implies the interval m through n, where m is not in the interval while n is (m, n) implies the interval m through n, where neither m and n are in the interval

Here is an example that uses an integer interval.

[0, 5] implies 0, 1, 2, 3, 4, 5 [0, 5) implies 0, 1, 2, 3, 4 (0, 5] implies 1, 2, 3, 4, 5 (0, 5) implies 1, 2, 3, 4 If w falls in the interval [0, 5], then 0 ≤ w ≤ 5 If x falls in the interval [0, 5), then 0 ≤ x < 5 If y falls in the interval (0, 5], then 0 < y ≤ 5 If z falls in the interval (0, 5), then 0 < z < 5

When graphed, a non-shaded circle is used for excluded values and either a shaded circle or an 'x' is used for included values.

[0, 5] [0, 5) (0, 5] (0, 5) x---------x x---------o o---------x o---------o 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

{TopOfPage} {101Science} {quiet} { TI-83} { Factors} {MathWords}

Domain and Range

The domain of a function are all the inputs for which a function is defined.
unknown functions and their respective domains ============================================== f(x) -- all real numbers (-∞, ∞) R: -∞ < x < ∞ g(x) -- all integers (-∞, ∞) Z: -∞ < x < ∞ h(x) -- all positive integers including zero [0, ∞) Z: x ≥ 0 i(x) -- all real numbers between 0 and 1, exclusive (0,1) R: 0 < x < 1 j(x) -- all real numbers between 0 and 1, inclusive [0,1] R: 0 ≤ x ≤ 1 k(x) -- all negative real numbers (-∞ 0) R: x < 0 l(x) -- all positive integers less than or equal to 10 (0, 10] Z: 0 < x ≤ 10 R implies real number Z implies integer

Domain values are the indepedent values.

The range of a function are all the outputs produced by the fuction.

Range values are the dependent values. The output of a function depends on its input; therefore, the range of a function depends on its domain.

{TopOfPage} {101Science} {quiet} { TI-83} { Factors} {MathWords}

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.

{TopOfPage} {101Science} {quiet} { TI-83} { Factors} {MathWords}

Home Previous Next