Chapter 1 Section 3
Prerequisites for Calculus

Sets of ordered pairs are called relations.
Certain relations are called functions.
(D) - domain consists of all x values in a relation.
(R) - range consists of all y values in a relation.

Function is a relation that assigns a single element of R to each element of D.

Vertical Line test of a function:
Any vertical line drawn can not touch the graph more than once

Real-Valued Functions of a Real Variable:
y - dependent variable - range
y - independent variable - domain
y = f(x)
y is a function of f.

Identifying Domain and Range:
- can not divide by zero
- can not have negative square root
 

Even or Odd Functions:
Even if f(-x) = f(x)
Odd if f(-x) = -f(x)
Examples:
 f(x) = x2 then (-x)2 = x2 therefore it is even
f(x) = x3 then (-x)3= -x3  therefore it is odd
f(x) = x+1 then (-x+1) = -x+1 therefore it is neither

Absolute Value Functions:
f(x) = |x+2|+|x-5|
Work it out algebraically: changing points x= -2 and x = 5
for x < -2:    x+2 < 0      x-5 < 0
                    f(x) = -(x+2) - (x-5)
                                -x-2-x+5
                    f(x) = -2x + 4

for  -2£x£5    x+2 ³ 0     x-5 £ 0
                    f(x) = x+2-(x-5)
                                x+2-x+5
                    f(x) = 7

for x > 5     x+2 > 0        x-5 > 0
                    f(x) = x+2+x-5
                    f(x) = 2x-3

therefore  f(x)  = {

-2x+4    (x<-2)
7          (-2£x £ 5)
2x-3      (x>5)

 

Sum, Difference, Products, and Quotients of functions:

f(x) = x+5    g(x) = x2 + 1

f+g f(x)+g(x) x+5+x2+1 = x2+x+6
f-g f(x)-g(x) x+5-(x2+1) = -x2+x+4
f*g f(x)*g(x) (x+5)(x2+1) = x3+5x2+x+5
f/g f(x)/g(x) (x+5)/(x2+1)

Composition of Functions:

f(x) = x-9    g(x) = x2 + 2

        Find f(g(2)).        g(2)=6 then f(6) = -3
        Find g(f(-4)).       f(-4)=-15 then g(-15) = 227
        Find f(g(x)).          x2 + 2 - 9
        Find g(f(x)).        (x-9)2 + 2