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 output values for the dependent variable (usually y), resulting from members of the domain being plugged into the function.

Explicit and Implicit Form Given that y is represents f(x) for a function. Explicit form is solved for y. example: Implicit form shows the relationship between y and x but is not solved explicitly for y.

Function Notation

Example: Lets take a closer look at this!

Example: An example of this type is referred to as a difference quotient. The difference quotient will be used extensively in this class!

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

Domain Restrictions: Denominator can not be zero Radicand of an even root has to be greater than or equal to zero. Argument of a logarithm must be positive. It is extremely important to consider domain restrictions throughout Calculus!

Examples Find the domain and range of each function:

Graph of a Function How do you tell if a graph is a function? One input value causes one output value. Vertical line test No Yes Yes

Basic Functions: Square root function: Identity function: Squaring function: Cubing function: Absolute value function: Rational function: Sine function: Cosine function:

Basic Transformations of y=f(x) c>0 y=f(x-c) f(x) shifted right c units. y=f(x+c) f(x) shifted left c units. y=f(x)-c f(x) shifted down c units. y=f(x)+c f(x) shifted up c units. y=-f(x) f(x) reflected about x. y=f(-x) f(x) reflected about y. y=-f(-x) f(x) reflected about the origin.

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

Polynomial Functions Degree: the highest power on a polynomial, n, right and left behavior depends on whether degree is odd or even. Lead coefficient: Coefficient of the highest degreed polynomial term; denoted an. Constant: Coefficient of the zero degreed polynomial term. 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 given an odd or even degree. 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 Evaluate each expression below stating the domain and range of each:

Composite Functions f◦g(x)=f(g(x)) The domain of f(g(x)) is any x value such that both of the following statements are true: - x is in the domain of g. and - g(x) is in the domain of f. In short: look at the domain of the simplified composite function AND of the function used as input into the composite function.

Examples Evaluate and state the domain of the composite function .

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. Example: f(x)=x2+x+2 f(-x)=(-x)2+(-x)+2= x2 – x + 2 NOT EVEN -f(-x)=-((-x)2+(-x)+2)= - x2 + x – 2 NOT ODD The function is neither even or odd.

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