MA 22000, Lesson 2 Functions & Addition/Subtraction Polynomials Algebra section of text: Sections 3.5 and 5.2, Calculus section of text: Section R.

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1 MA 000, Lesson Functions & Addition/Subtraction Polynomials Algebra section of tet: Sections.5 and 5., Calculus section of tet: Section R.1 Definition: A relation is any set of ordered pairs. The set of first components (input values) in the ordered pairs is called the domain of the relation. The set of second components (output values) is called the range of the relation. (Sometimes the domain is referred to as the set of independent variables and the range is referred to as the set of dependent variables.) Relations may be represented as sets, tables, diagrams (mappings), graphs, or equations. Definition: A function is a relation in which, for each value of the first component of the ordered pairs, there is eactly one value of the second component. (In other words, a function is a relation in which no two ordered pairs have the same first components but different second components.) Each element or component of the domain (input values) is paired to one and only one element or component of the range (output values). Eample 1: Is each relation represented a function? Give the domain and range. a) {(9,81), (4,16), (5,5), (,4), ( 6,6)} Is this set a function? b) y -1 0 Is this table a function? c) 5 - DOMAIN 1-1 RANGE Is this mapping a function? d) Is this graph a function? (1,-1) (,-1) 1 (-,-)

2 Vertical Line Test: If any possible vertical line intersects the graph of a relation in no more than one point, then the relation is a function. (See page 04 of section.5 figure 44.) e) y Does this equation represent a function? There are other eamples about relations/functions and domain/range found in the tetbook, section.5 of the first part of the tetbook, pages Function Notation: Functions can be named by using letters. This name can be used to write the function. For eample; the function h represented by y can be written as h( ) domain (the ) is inside the parentheses and the result (the y) is the matching element of the range. The notation f( ) simply represents the y or dependent variable. It is read f of or function of.. This type of notation is known as function notation. The element of the Eample : Given the function g( ) 1, find the following function values. a) g( ) b) g(0) c) g(4) d) g( ) Eample : Given the functions a) f ( ) b) F( a 1) c) f ( r ) d) 1 F b e) f ( ) f () f ( ) and F( ) 1, find

3 A linear function is defined by f ( ) a b for real numbers a and b. The value of a is the slope m and the value of b is the y-intercept. A function may be defined by more than one equation, such as the piecewise-defined function in eample 4. Eample 4: Given this piecewise-defined function, find the following. if 1 f( ) if 1 a) f(0) b) f(1) c) f ( ) if 1 In most functions the implied domain is the set of all real numbers (inputs) that yield real number values for the function values (outputs). In real life problems, the domain and range are only the values that are reasonable. Eample 5: Find the domain of each function. a) f ( ) b) G( ) 4 c) d( t) 50 t, where d is the distance in miles traveled by a car in t hours Definition: A polynomial is a term (a number, a variable, or the product or quotient of a number and one or more variables raised to a power) or a finite sum/difference of terms in which all variables have whole number eponents and no variables appear in denominators. A polynomial k in a single variable is an algebraic epression whose terms are all of the form a, where a (the coefficient) is any real number and k (the eponent) is a nonnegative integer. The degree of the polynomial is the greatest eponent in the polynomial and the coefficient of the term with that greatest eponent is called the leading coefficient. Any number term (without a variable) is

4 called the constant term or constant. Many polynomial answers are written in descending powers of the variable. Definitions: A polynomial that is written in order of descending powers of the variable is said to be in standard form. A polynomial with only one term is a monomial. A polynomial with two unlike terms is called a binomial. A polynomial with three unlike terms is called a trinomial. Polynomial Standard Form Degree Leading Coefficient a 1a a 6 5 1a a 4a To add two or more polynomials, simply combine like terms. To subtract a polynomial, add the opposite (distribute the negative sign). The book demonstrates how to add or subtract horizontally or vertically. The key is to remember to distribute the negative (minus) through each term of the polynomial that is subtracted. Eample 6: Add or subtract or combine where possible in each polynomial epression. a ) ( 9) (5 10) b) (4 ) (1 9) (5 7 ) c a a a a a a ) ( 4 1) [( ) (4 9 7)] The position function gives the height above ground of an object falling to earth or hurled upward then falling back to earth. It is h 16t v0t s0. It is a function of time and the height will be in feet. Velocity is in feet per second and the initial height will be in feet, time in seconds. The coefficient of t, the v 0 is the initial velocity (velocity at time 0). The s 0 will be the initial height of the object (height at time 0). Note: If the initial velocity is positive, the object is projected upward; if negative, the object was projected downward; if the initial velocity is zero, the object is simply dropped. 4

5 Eample 7: An object has the position function h( t) 16t 48t 6. Eplain what the 48 and the 6 represent. Find and interpret the following. a) h(0) b) h() c) h() d) h(5) Eample 8: The four sides of a quadrilateral can be represented by a 5, a 1, 5a 6, and 4a. Find a polynomial that represents the perimeter of the quadrilateral. If a = 1, what is the perimeter? Eample 9: A manufacturer can produce and sell gadgets per week. The total cost of producing these gadgets (in dollars) is given by C( ) The revenue from the sale of these gadgets (in dollars) is R( ) 6.6. Find a function to represent the profit from the production and sell of gadgets per week. What is the profit from the sale of 5 gadgets? 5