P.3 Functions and Their Graphs

Objectives Use function notation to represent and evaluate a function. Find the domain and range of a function. Sketch the graph of a function. Identify different types of transformations of functions. Classify functions and recognize combinations of functions.

Function A function must make an assignment to each number in the domain. A function can assign only one number to any given number in the domain. function not a function

Domain and Range Domain: the set of values that the independent variable (usually x) can take on Range: the set of values for the dependent variable (usually y) if all numbers in the domain are plugged into the function

Implicit Form Not explicitly solved for the variable Explicit form

Function Notation

Example

Example

Domain and Range Domain is explicitly defined D: [4,5] R: [1/21, 1/12] D: x≠±2 (implied) R: f(x)≠0

Domain Restrictions Denominator can not be zero Radicand of an even root has to be greater than or equal to zero.

Examples Find the domain and range.

Graph of a Function How do you tell if a graph is a function? Vertical line test No Yes Yes

Basic Functions Look at the graphs on page 22. y=x y=x2 y=x3 y=√x y=sinx y=cosx

Basic Transformations (p.23) y=f(x) y=f(x-c) y=f(x+c) y=f(x)-c y=f(x)+c y=-f(x) y=f(-x) y=-f(-x) Shifts right Shifts left Shifts down Shifts up Reflects about x-axis Reflects about y-axis Reflects about origin

Elementary Functions Algebraic (polynomial, radical, rational) Trigonometric Exponential and logarithmic

Polynomial Functions Although a graph of a polynomial function can have several turns, eventually the graph will rise or fall without bound as x moves to the right or left.

Polynomial Functions an(leading coefficient): determines right or left behavior of the graph an>0 an<0 EVEN degree ODD degree

Combinations and Domains sum: f(x)+g(x) what's in both f and g difference: f(x)-g(x) product: f(x)g(x) quotient: f(x)/g(x) (as long as g(x)≠0)

Example f(x)=2x-3 and g(x)=√x+1 D: (-∞,∞) D: [-1,∞) f(x)+g(x)=2x-3+√x+1 f(x)-g(x)=2x – 3 – √x+1 f(x)g(x)=(2x – 3)(√x+1) f(x)/g(x)=(2x – 3)/(√x+1) D:[-1,∞) D:(-1,∞)

Composite Functions f◦g(x)=f(g(x)) g◦f(x)=g(f(x)) Domain: look at the final answer AND any steps before simplifying

Examples f(x)=√x and g(x)=x2+1 Find the domains of the composite functions.

Even and Odd Functions f(-x)=f(x) even (symmetric to y-axis) f(-x)=-f(x) odd (symmetric to origin) Symmetric to x-axis? If it is a polynomial, you can just look at the powers of x.

Examples f(x)=x3-x f(-x)=(-x)3-(-x) =-x3+x =-f(x) odd (symmetric to origin) g(x)=1+cosx g(-x)=1+cos(-x) =1+cosx=g(x) even (symmetric to y-axis)

Neither Even or Odd Of course, some functions are neither even or odd. f(x)=x2+x+2 f(-x)=(-x)2+(-x)+2=x2 – x + 2

Homework P.3 (page 27) #3, 9, 27-31 odd 32, 33-37 odd 41-45 odd 61, 63, 69, 71