Chapter Six Functions

Lesson 1 – Relations, Functions – notation, domain & range Lesson 2 – Symmetry – even & odd functions Lesson 3 – Composite Functions – f(g(x)) Lesson 4 – Inverse Functions – f -1 (x), switch coordinates Lesson 5 – Compound (piecewise) Functions Lesson 6 – Graphs of Functions Lesson 7 – Transformations of Graphs

A function is a dependant relationship between two numbers and a rule which assigns a unique output value to each input value 7 49 x2x2

A function is a dependant relationship between two numbers and a rule which assigns a unique element of range (y) to each element of the domain (x) 7 49 x2x2 Domain (inputs) Range (outputs) Rule x f(x) y 

Function notation makes clear which variable is independent and which is dependant Function Notation y = x 2, normal equation notation f (x) = x 2, general function notation C( b) = rule, typical usage Cost, dependant # books printed independent (in parenthesis)

Representations of Functions Rule or Equation f (x) = 3x Ordered Pairs (1, 3) (5, 15) (x, 3x) Table of values x13x f(x)3153x Graph

Skills Today Ex. 1 Write each equation as a function of x y = 2x – 4 f(x) = 2x – 4 y = |x + 3| – 5 y = 9x 2 – 25 f(x) = |x + 3| – 5 f(x) = 9x 2 – 25 Ex. 2 Evaluate each function for the given value of x f(6) = 2x – 4 f( – 9 ) = |x + 3| – 5 f( 5 / 3 ) = 9x 2 – 25 f(6) = 2(6) – 4 f( – 9 ) = |(– 9) + 3| – 5 f( 5 / 3 ) = 9( 5 / 3 ) 2 – 25 f(6) = 8 f(– 9) = 1 f( 5 / 3 ) = 0 f(3) = x 2 + 5x + 1f(3) = (3) 2 + 5(3) + 1 f(3) = 25

Ex. # 3 – Use vertical line test to verify a relation is a function. A function is a dependant relationship between two numbers and a rule which assigns a unique output value to each input value

Vertical Line Test – sweep a vertical line across the graph, functions only contact graph at one point. FunctionNot a Function

Ex. 4 – Use the graph to determine the domain and range of the function D: All realsR: All reals D: |x| ≥ 2 D: |x| ≤ 2 D: x ≥ 0R: All realsR: y ≤ 0 R: 0 ≤ y ≤ 2

Finding Domain of a Function Use Graph Provided to you Story Problem dictates Values for which function is Defined

Provided to you f(x) = x 2 – 7 ; x > 0 f(x) = |x – 5| + 1 ; 2 < x < 9 Use Graph Story Problem dictates Traveling 45 mph, how far can you drive in t hours? D(t) = 45t We can see that time t is the domain. If just starting trip, only zero and positive times make sense. t ≥ 0 Values for which function is Defined We know that this function is undefined for the set of real numbers if x is negative. x ≥ 0 Finding Domain

Values for which function is Defined Most of our functions at this time are defined for all real numbers, and the domain is all reals 2 EXCEPTIONS Square Root Functions Rational Functions non-negative means ≥ 0 3x – 5 ≥ 0 solvex ≥ 5 / 3 x + 4 ≠ 0 solvex ≠ – 4

x + 3 ≥ 0x ≥ – 3 6 – x ≥ 0x ≤ 6 3x + 9 ≥ 0x ≥ – 3 x – 8 ≠ 0x ≠ x ≠ 0x ≠ – 1 x 2 – 1 ≠ 0  x 2 ≠ 1 x ≠  1

Basic Operations on Functions +, –, ×, ÷ f(x) = 2x + 1, g(x) = 5x, determine h(x) = f(x) + g(x) h(x) = (2x + 1) + 5x = 7x + 1 f(x) = 2x + 1, g(x) = 5x, determine h(x) = f(x) – g(x) h(x) = (2x + 1) – 5x = – 3x + 1 f(x) = 2x + 1, g(x) = 5x, determine h(x) = f(x) × g(x) f(x) = 2x + 1, g(x) = 5x, determine h(x) = f(x) ÷ g(x) h(x) = (2x + 1) × 5x = 10x 2 + 5x h(x) = (2x + 1) ÷ 5x = 2x + 1 5x

Symmetry y – axis symmetry Even Function Algebraic Test f(x) = f(-x) f(x) = x 4 – x 2 – 2 x 4 – x 2 – 2 = (-x) 4 – (-x) 2 – 2 x 4 – x 2 – 2 = x 4 – x 2 – 2 origin symmetry Odd Function Algebraic Test f(x) = - f(-x) f(x) = x 5 x 5 = - (-x) 5 x 5 = - -x 5 x 5 = x 5

Test for Symmetry Even Function Neither Even nor Odd Odd Function

h(x) = h(-x) x 2 – 3 = (-x) 2 – 3 x 2 – 3 = x 2 – 3 even function r(x) = - r(-x)h(x) = - h(-x) r(x) = r(-x) x 2 – 3 = -[(-x) 2 – 3] x 2 – 3 = -[x 2 – 3] x 2 – 3 = -x not odd function x = (-x) x = -x not even function x = -[(-x) 3 + 1] x = -[-x 3 + 1] x3 + 1 = x 3 - 1] not odd function, neither m(x) = m(-x) 1/x = 1/(-x) 1/x = -1/x not even function m(x) = - m(-x) 1/x = -[1/(-x)] 1/x = -[-1/x] 1/x = 1/x odd function

Composite Functions A composite function is the result of placing one function inside another at each variable. f(g(x)) Places the function g(x) inside the function f(x). f(x) = 2x – 5 g(x) = x f(g(x)) = 2(x 2 + 1) – 5 = 2x 2 – 3

f(x) = x 2 + x,g(x) = 1 / x, r(x) = x – 2 Writing Composite Functions r(f(x)) =( f(x) ) – 2= (x 2 + x) – 2 = x 2 + x – 2 f(r(x)) =( r(x) ) 2 + ( r(x) )= ( x – 2 ) 2 + ( x – 2 ) = x 2 – 3x + 2 g(f(x)) = f(g(r(x))) =

f(x) = x 2 + x,g(x) = 1 / x, r(x) = x – 2 Evaluating Composite Functions r(f(x)) = x 2 + x – 2 = (4) 2 + (4) – 2 = 18 f(r(4)) =f(2) = ( 2 ) 2 + ( 2 ) = 6 f(r(x)) = x 2 – 3x + 2 g(f(3)) = f(4) = (4) 2 + (4) = 20 r(20) = (20) – 2 = 18 r(f(4)) = r(4) = 4 – 2 = 2 = (4) 2 – 3(4) + 2 = 6 f(3) = (3)2 + (3) = 12

f(x) = x 2 + x, g(x) = 1 / x, r(x) = x – 2 Domain of Composite Functions r(f(x))= x 2 + x – 2 f(r(x))= x 2 – 3x + 2 g(f(x)) f(g(r(x))) all real’s x ≠ 0 all real’s Determine Domain of individual functions. Domain must meet conditions of inside functions and overall f(x), inside, no restrictions; overall, polynomial no restrictions, all real’s r(x), inside, no restrictions; overall, polynomial no restrictions, all real’s f(x), inside, no restrictions; overall, x 2 + x ≠ 0, x ≠ 0, – 1 r(x), inside, no restrictions; g(r(x)), x -2 ≠ 0; overall, same, x ≠ 2

Inverse Functions Inverse – 1. Reversed in order, nature, or effect. * * American Heritage Dictionary, 1978 Inverse operations: add – subtract, multiply – divide, square – square root Inverse functions f(x) = 3x + 2, and g(x) = (x – 2)/3 Evaluate f(x) at 12= 38then evaluate g(x) at 38 = 12 Also evaluate g(x) at 17= 5then evaluate f(x) at 5 = 17 Given two functions f(x) and g(x) such that f(g(x)) = x, and g(f(x)) = x, then f(x) & g(x) are inverses of each other. f -1 (x) = g(x), g -1 (x) = f(x)

Properties of Inverse functions Inverse – 1. Reversed in order, nature, or effect. * In Inverse functions the x and y are switched. Rule or Equation f (x) = 3x, y = 3x x13x f(x)3153x Graph Ordered Pairs (1, 3) (5, 15) (x, 3x) Table of values x = 3y f -1 (x) = ? Ordered Pairs (3, 1) (15, 5) (3x, x) f(x)3153x x15x

Graphing Inverses w/symmetry Previous slide y = 3x and inverse x = 3y The graph of a function and its inverse are symmetrical about the line y = x

Graphing Inverses w/symmetry Ex. Graph the inverse of f(x) = 10 x Ex. Graph the inverse of g(x) = tan x; |x| < π/2 y = x

Writing Inverse Functions Ex. Write the inverse of (h)x = x 3 – 5, h -1 (x) = Inverse – 1. Reversed in order, nature, or effect. * 1. Function notation h(x) is not convenient, switch to y. y = x Switch the x and y to write inverse. x = y Solve for y when possible to obtain h -1 (x). 4. Write in function notation with inverse h -1 (x) for y.

Testing or Determining if functions are inverses Given two functions f(x) and g(x) such that f(g(x)) = x, and g(f(x)) = x, then f(x) & g(x) are inverses of each other. Use the definition with the given functions, your work is the answer the result in each case should be x Ex. r(x) = x / 2 + 1, t(x) = 2x – 2 r(t(x)) =t(r(x)) = Ex. m(x) =x 2 – 1 m(n(x))=n(m(x)) = Each case resulted in x, any other result and functions are not inverses.