Functions & Their Graphs (P3)

Functions & Their Graphs (P3)

September 10th, 2012September 10th, 2012

I. Functions & Function Notation

Ex. 1: For the function f defined by f(x) = 3x2 - 4x, evaluate each expression.a. f(-1)b. f(3a)c. f(b+2)d. f (x + Δx)− f (x)

Δx

You try:

For the function f defined by f(x)=2x+4, evaluate f (x +Δx)− f(x)

Δx

The Domain & Range of a Function

Def. The domain is the set of all input values for x. The range is the set of all outcomes of f(x).

Ex. 2: Find the domain and range of each function.a. f(x)=x2+2b. g(x)= 4x+1

c. h(t)=sec t

d. f (x)=1

3x−1

III. The Graph of a Function

Def: An equation represents a function if for each x-value, there only exists one corresponding y-value, or it passes the vertical line test.

Ex. 3: Determine whether y is a function of x.a. x2 + y2 =4

b. y+ x2 =4

IV. Transformations of FunctionsBasic Transformations (c>0):Original Graph y=f(x)Horizontal shift c units right y=f(x-c)Horizontal shift c units left y=f(x+c)Vertical shift c units up y=f(x)+cVertical shift c units down y=f(x)-cReflection about the x-axis y=-f(x)Reflection about the y-axis y=f(-x)Reflection about the origin y=-f(-x)

Ex. 4: Describe each transformation, then use your description to write an equation for each graph.a.

b.

c.

d.

e.

V. Classifications & Combinations of Functions

Def: The composite of function f with function g is given by ( f og)(x)= f(g(x))

The domain of f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f.Ex. 5: Given and , findeach composite function.

f (x)=4x

g(x)=x2 −1

a. f og

b. go f

You Try: Given and , findeach composite function.

f (x)=x3 +1 g(x)=2x−6

a. f og

b. go f

c. go f (−4)

Def: A function y=f(x) is even if f(-x)=f(x). It is odd if f(-x)=-f(x). Even functions are symmetric about the y-axis, odd functions are symmetric about the origin.

Ex. 6: Determine whether each function is even, odd, or neither. Then use a graphing utility to verify your result.a.

f (x)=x3(x−4)

b. f (x)=xsinx

c. f (x)= x5