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OCF.02.1 - Functions: Concepts and NotationsOCF.02.1 - Functions: Concepts and Notations

MCR3U - SantowskiMCR3U - Santowski

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(A) Concept of Functions & Relations(A) Concept of Functions & Relations

In many subject areas, we see relationships that In many subject areas, we see relationships that exist between one quantity and another quantity.exist between one quantity and another quantity.

– ex. Galileo found that the distance an object falls is ex. Galileo found that the distance an object falls is related to the time it falls.related to the time it falls.

– ex. distance travelled in car is related to its speed.ex. distance travelled in car is related to its speed.– ex. the amount of product you sell is related to the ex. the amount of product you sell is related to the

price you charge.price you charge.

All these relationships are classified All these relationships are classified

mathematically as mathematically as RelationsRelations..

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(B) Representation of Functions & Relations(B) Representation of Functions & Relations

Relations can be expressed using Relations can be expressed using many methodsmany methods..

   ex. table of valuesex. table of values

TimeTime 00 11 22 33 44 55

Distance Distance 00 55 2020 4545 8080 125125

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(B) Representation of Functions & Relations(B) Representation of Functions & Relations

Relations can be expressed using graphsRelations can be expressed using graphs

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(B) Representation of Functions & Relations(B) Representation of Functions & Relations

Relations can be expressed using ordered pairs i.e. (0,0), Relations can be expressed using ordered pairs i.e. (0,0), (1,5), (2,20), (3,45), (4,80), (5,125)(1,5), (2,20), (3,45), (4,80), (5,125)

The relationships that exist between numbers are also The relationships that exist between numbers are also expressed as equations: s = 5texpressed as equations: s = 5t22

This equation can then be graphed as follows:This equation can then be graphed as follows:

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(C) Terminology of Functions & Relations(C) Terminology of Functions & Relations

Two terms that we use to describe the relations Two terms that we use to describe the relations

are are domaindomain and and rangerange..

DomainDomain refers to the set of all the first refers to the set of all the first elements, input values, independent variable, elements, input values, independent variable, etc.. of a relation, in this case the timeetc.. of a relation, in this case the time

  

RangeRange refers to the set of all the second refers to the set of all the second elements, output values, dependent values, etc... elements, output values, dependent values, etc... of the relation, in this case the distance.of the relation, in this case the distance.

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(D) Functions - The Concept(D) Functions - The Concept

A A functionfunction is a special relation in which each is a special relation in which each singlesingle domain element domain element corresponds to exactly corresponds to exactly oneone range element. In other words, each input range element. In other words, each input value produces one unique output valuevalue produces one unique output value

   ex. Graph the relations defined by y = 2xex. Graph the relations defined by y = 2x22 + 1 and x = 2y + 1 and x = 2y22 + 1  + 1 

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(D) Functions - The Concept(D) Functions - The Concept

Q? In what ways do the two graphs differ?Q? In what ways do the two graphs differ?

   In the graph of y = 2xIn the graph of y = 2x22 + 1, notice that each value of x has + 1, notice that each value of x has

one and only one corresponding value of y.one and only one corresponding value of y.

   In the graph of x = 2yIn the graph of x = 2y22 + 1, notice that each value of x has + 1, notice that each value of x has

two corresponding values of y.two corresponding values of y.

   We therefore distinguish between the two different kinds of We therefore distinguish between the two different kinds of

relations by defining one of them as a function. So a relations by defining one of them as a function. So a function is special relation such that each value of x has function is special relation such that each value of x has one and only one value of y.one and only one value of y.

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(E) Functions - Vertical Line Test(E) Functions - Vertical Line Test

To determine whether or not a relation is in fact a To determine whether or not a relation is in fact a function, we can draw a vertical line through the function, we can draw a vertical line through the graph of the relation. graph of the relation.

If the vertical line intersects the graph more than If the vertical line intersects the graph more than once, then that means the graph of the relation is not once, then that means the graph of the relation is not a function. a function.

If the vertical line intersects the graph once then the If the vertical line intersects the graph once then the graph shows that the relation is a function.graph shows that the relation is a function.

See the diagram on the next slideSee the diagram on the next slide

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(F) Functions - Vertical Line Test(F) Functions - Vertical Line Test

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(G) Functions - the Notation F(x)(G) Functions - the Notation F(x)

So far, we have written equations in the form y = 2x + 5 or y = 3xSo far, we have written equations in the form y = 2x + 5 or y = 3x22 - 4. - 4.

These equations describe the relationship between x and y, and so they These equations describe the relationship between x and y, and so they describe relationsdescribe relations

Because these two relations are one-to-one, they are also functionsBecause these two relations are one-to-one, they are also functions

Therefore we have another notation or method of writing these equations Therefore we have another notation or method of writing these equations of functions. We can rewrite y = 2x + 5 as f(x) = 2x + 5. of functions. We can rewrite y = 2x + 5 as f(x) = 2x + 5.

This means that f is a function in the variable x such that it equals 2 times This means that f is a function in the variable x such that it equals 2 times x and then add 5. x and then add 5.

We can rewrite y = 3xWe can rewrite y = 3x22 - 4 as g(x) = 3x - 4 as g(x) = 3x22 - 4. - 4.

This means that g is a function in the variable x such that it equals 3 times This means that g is a function in the variable x such that it equals 3 times the square of x and then subtract four.the square of x and then subtract four.

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(G) Internet Review(G) Internet Review

We will now go to an internet lesson, We will now go to an internet lesson, complete with explanations and complete with explanations and interactive applets to review the key interactive applets to review the key concepts about functions:concepts about functions:

Functions from Visual CalculusFunctions from Visual Calculus

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(H) Working with f(x)(H) Working with f(x)

For the function defined by b(t) = 3tFor the function defined by b(t) = 3t22 - t + 3, evaluate b(4): - t + 3, evaluate b(4):

b(4) = 3(4)b(4) = 3(4)22 – (4) + 3 = 48 – 4 + 3 = 47 – (4) + 3 = 48 – 4 + 3 = 47

So notice that t = 4 is the “input” value (or the value of So notice that t = 4 is the “input” value (or the value of independent variable) and 48 is the “output” value (or the independent variable) and 48 is the “output” value (or the value of the dependent variable)value of the dependent variable)

So we can write b(4) = 48 or in other words, 48 (or b(4)) is So we can write b(4) = 48 or in other words, 48 (or b(4)) is the “y value” or the “y co-ordinate” on a graphthe “y value” or the “y co-ordinate” on a graph

So we would have the point (4,48) on a graph of t vs b(t)So we would have the point (4,48) on a graph of t vs b(t)

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(H) Working with f(x)(H) Working with f(x)

ex. For the function defined by b(t) = 3tex. For the function defined by b(t) = 3t22 - t + 3, find: - t + 3, find:

(a) b(-2) (a) b(-2) (b) b(0.5)(b) b(0.5) (c) b(2)(c) b(2) (d) b(t - 2)(d) b(t - 2) (e) b(t(e) b(t22)) (g) b(1/x)(g) b(1/x)  

ex. For the function defined by d(s) = 5/s + sex. For the function defined by d(s) = 5/s + s22, find, find

(a) d(0.25) (b) d(4)(a) d(0.25) (b) d(4) (c) d(-3)(c) d(-3)  

ex. For the function defined by w(a) = 4a - 6, find the value ex. For the function defined by w(a) = 4a - 6, find the value of of aa such that w(a) = 8 such that w(a) = 8

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(I) Internet Links(I) Internet Links

College Algebra Tutorial on IntroductiCollege Algebra Tutorial on Introduction to Functions - West Texas A&Mon to Functions - West Texas A&M

Functions Lesson - I from Functions Lesson - I from PurpleMathPurpleMath

Functions Lesson - Domain and RangFunctions Lesson - Domain and Range from e from PurpleMathPurpleMath

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(J) Homework(J) Homework

Nelson Text , page 234-237, Q1-8 together in class, Nelson Text , page 234-237, Q1-8 together in class, Q9,11,13,16,17,18-21 are word problemsQ9,11,13,16,17,18-21 are word problems