Functions

Venn Diagram of the Real Number System

Subsets of Real Numbers Label the following on your diagram integers (Z) natural numbers (N) irrational numbers (I) real numbers (R). whole numbers (W) rational numbers (Q)

Place the following numbers in your diagram. -3 23.4 13 ¼ Π .2222…. √3

Set-Builder Notation x > 5 becomes {x | x > 5, x Є R} {3,2,1,0,-1,-2…} becomes {x | x≤3, x Є Z} x is a multiple of 5 becomes {x| x = 5n, n Є Z}

Interval Notation x > 2 is equivalent to (2, ∞) this is considered an open interval -1≤ x ≤ 3 is equivalent to [-1, 3] this is considered a closed interval x > 5 or x ≤ 2 is equivalent to (-∞, 2] U (5,∞) Can you graph each of these on a number line?

Relations A relation is a rule (mathematical or otherwise) that relates two quantities

What is a function? A function is a RELATION which pairs each input value with EXACTLY one output value.

Think of a box. Input Output x domain independent variable y range f(x)

Function or not? a) (1,2) (1,3) (1,4) (2,5) no b) (2,1) (3,1) (4,1) (5,2) yes See what these look like in a mapping diagram, a table of values, and a graph. What is any easy way to tell whether a graph represents a function? vertical line test

Function or not? Input a person, output that person’s birthday Input x, output 2x – 1 Input a perosn, output current math grade Input a person, output their biological mother Input x, output solutions to x = y2 Input a woman, output her children The last two relations are not functions. A function can only give you one “answer” for a particular input.

Function or not? Is y a function of x? (Hint: use both the graph and the equation.) 2x + 3y = 7 yes y + 3= x2 x2 + y2 = 25 no

Evaluating Functions f(x) = 2x - 5 Find f(-5), f(2x), and f(x + 1) 13, 9x2 – 3, x2 + 4x + 1 f(x) = x2 + 2x Find f(-3) and f(x - 1) 3, x2 - 1

Determining Domain 1. f(x) = 2x + 3 2. a) f(x) = 1 ∕ x b) f(x) = 1 ∕ (x + 3) c) f(x) = 1 ∕ (x2 + x - 12) 3. a) f(x) = √ x b) f(x) = √(10-2x) c) f(x) = √(x2 + 25) d) f(x) = √(x2 - 16)

What about this one?  

Piecewise Functions (Part II) f(x) = { 2 if x< 0 { 2 +x if x ≥ 0 Evaluate f(-4) and f(7). Graph this. f(-4) = 2 f(7) = 9

The speed of a particular vehicle in mph can be represented by the following piecewise function when t is the time in seconds. Find the speed of the vehicle at the given times. { 4t if 0 ≤ t ≤15 v(t) = { 60 if 15 < t < 240 { 1500 – 6t if 240 ≤ t ≤ 250 a) v(5) b) v(15) c) v(245) Describe the speed of the vehicle over time. Interpret the v(t) notation and your results. a) 20 b) 60 c) 30

Sometimes the domain is implied by the context of the application Sometimes the domain is implied by the context of the application. For instance, in A(r) = πr2, we assume r > 0. Give a formula for the area of a square as function of its perimeter. A(P) = P2/16

Difference Quotients (Part III) f (x + h) – f(x) ; h ≠ 0 h A formula used frequently in calculus. See diagram for how this relates to slope. For f(x) = x2 + 3x, evaluate f(x + h) – f(x) .