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