1.2 Functions fguilbert orrhs

-To identify a function. -To determine domain & range Objective: -To identify a function. -To determine domain & range -To graph functions fguilbert orrhs

What is a relation ? A relation in math is a group of ordered pairs. { ( 4, 5), (6,7), (9,10)} { (1,1), (1,2), (1,2) } fguilbert orrhs

A function is a relation in which no two ordered pairs have the same x-value. {(3,2), (4,5), (6,-1), (0,2)} is a function because all the x-values are different. fguilbert orrhs

D R x y A function is the set of (x,y) such book definition D R x y A function is the set of (x,y) such that X is an element of the domain D and Y is the corresponding element of the range R. fguilbert orrhs

This is formally written { (x,y) l y = f(x) } D R x maps onto y y x This is formally written { (x,y) l y = f(x) } the set such that fguilbert orrhs

A function -no two ordered pairs have the same x-value. What’s NOT a Function? {(-3,2) ,(4,5) , (6,-1) ,(4,2)} is NOT a function. fguilbert orrhs

Range - set of y values. {(3,2) , (4,5) , (6,-1) , (0,2)} Domain - set of x values. Range - set of y values. {(3,2) , (4,5) , (6,-1) , (0,2)} Domain {0,3,4,6} Range {-1,2,5} Arrange order. Do not repeat numbers. fguilbert orrhs

Range - set of y values. {(3,2) , (4,5) , (6,-1) , (0,2)} Domain - set of x values. Range - set of y values. {(3,2) , (4,5) , (6,-1) , (0,2)} Domain {0,3,4,6} Range {-1,2,5} { } means the set of fguilbert orrhs

A vertical line only intersects 1 point. What is a Function? A vertical line only intersects 1 point. Y X passes the vertical line test fguilbert orrhs

What’s NOT a Function? If the vertical line intersects 2 points, the graph is NOT a function. Y X Fails the vertical line test fguilbert orrhs

The value of y depends on x What’s a Function The value of y depends on x x function Output y fguilbert orrhs

Y is the Dependent variable. Since Y DEPENDS on X, Y is the Dependent variable. X is the Independent variable fguilbert orrhs

same as f(x) = 3x +2 read: the function of X Let’s let y = 3x + 2 The number you substitute for X determines the number you get for Y. same as f(x) = 3x +2 read: the function of X fguilbert orrhs

Can also be written as g(u) = 3u + 2 Read g(u)= The function g of u = Function Notation y = 3u + 2 Can also be written as g(u) = 3u + 2 Read g(u)= The function g of u = fguilbert orrhs

Function Notation (x,y) is a solution of y = 3x + 2. (x, f(x)) is a solution of f(x) = 3x + 2. f(x) is the same thing as y. fguilbert orrhs

The bike’s height is a function of what ? fguilbert orrhs

Function Notation f(1) = 3(1) + 2 f(1) = 3 + 2 f(1) = 5 If x = 1 and f(x) = 3x + 2, then: f(1) = 3(1) + 2 f(1) = 3 + 2 f(1) = 5 f (1) means use 1 for x Remember fguilbert orrhs

h(x)= - 2x2 -x Find h(-3) h(-3)= - 2(-3)2 - ( -3) h(-3)= - 2(9) +3 Try this one h(x)= - 2x2 -x Find h(-3) h(-3)= - 2(-3)2 - ( -3) h(-3)= - 2(9) +3 h(-3)= -15 fguilbert orrhs

Function Notation f(x) = 3x2 + 2x f(x+h) = 3(x+h)2 + 2(x+h) f(x+h) = 3(x2+2xh +h2) + 2(x+h) = 3x2+ 6xh +3h2 +2x +2h Be careful fguilbert orrhs

f(x) = 3 X g(x) = x+1 find f ( g (x)) Main function is f(x) f(x)= 3 X = 3 ( ) What replaces X? (x+1) f(g(x)= 3x + 3 fguilbert orrhs

f(x) = X 2 g(x) = x-2 find f ( g (x)) f(x)= X 2 Main function is f(x) f(x)= X 2 = ( ) 2 What will replace X? (x-2) 2 = (x-2)(x-2) = x2 -4x + 4 fguilbert orrhs

Finding the domain and range What x values can be used in the function f(x) = 2x + 3 ? Any real number. So D = set of reals. fguilbert orrhs

What is the result for the y values? They are also real numbers. f(x)= 2x+3 What is the result for the y values? They are also real numbers. So R = set of reals. fguilbert orrhs

What x values can be used in the function f(x) = 3 / x ? Any real number except 0 can be used. So D = set of reals, x = 0. What set will result for the y values? These are also real numbers except 0. So R = set of reals, y = 0. fguilbert orrhs

What x values can be used in the function f(x) = √ x ? Only + reals and 0 can be used. Why? So D = {x > 0 } . What set will result for the y values? These are also 0 or larger. So R = { y > 0 } . fguilbert orrhs

Find the domain and range x - 3 = 0 D = {reals x = 3} R = {reals y = 0} x + 2 > 0 D = { x > - 2 } R = { y < 0 } f(x) = _4_ x - 3 f(x) = -√ x + 2 fguilbert orrhs

Find domain and range y = x2 - 2 D = all real nos. R > -2 fguilbert orrhs

Find Domain and Range Domain -4 < x < 3 fguilbert orrhs

Find Domain and Range Range -1 < y < 2 fguilbert orrhs

D = { x l -3< x < 4 } Piecewise Function Find domain x values are from -3 & 4 -3< X < 4 D = { x l -3< x < 4 } fguilbert orrhs

Piecewise Function Find range Y values are -1, and 1 to 4 y = -1 or 1< y< 4 fguilbert orrhs

x - 4 < x < -1 f(x)= x2 x > 0 2 different equations Graph Domain x - 4 < x < -1 x2 x > 0 f(x)= 2 different equations fguilbert orrhs

x - 4 < x < -1 x2 x > 0 f(x)= Graph one at a time. f(x) = x domain x - 4 < x < -1 x2 x > 0 f(x)= Graph one at a time. f(x) = x fguilbert orrhs

x - 4 < x < -1 x2 x > 0 f(x)= f(x) = x2 Graph domain fguilbert orrhs

Determination can conquer almost anything. Thought for the day Determination can conquer almost anything. fguilbert orrhs