Section 1.1: Four Ways to Represent a Function

v Functions arise whenever one quality depends on another. ie. (x,y)

• A function f is a rule that assigns to each element x in a Set A exactly one element y, called f(x), in a Set B.
• Set A (x) is the Domain/Independent variable of the function
• Set B (y) is the Range/Dependent Variable

Four Ways to Represent a Function:

• Verbally= A Description in Words
• Numerically= Table of Values
• Visually= Graph
• Algebraically=Explicit Formula

-The Vertical Line Test: A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

Example:

A rectangular storage container with an open top has a volume of 10 m³. The length of the base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base

Solution:

The area of the base is (2w)w=2w², so the cost in dollars, of the material for the base is 10(2w²). Two of the sides have area wh and the other two have area 2wh, so the cost of the material for the sides is 6[2(wh) + 2(2wh)]. The total cost is therefore:

C=10(2w²)+ 6[2(wh) + 2(2wh)]= 20w² + 36wh

To express C as a function of w alone, we need to eliminate h and we do so by using the fact that the volume is 10 m³. Thus

w(2w)h = 10

h =

Substituting this into the expression for C, we have

C=20w² + 36w() = 20w² + ()

Therefore, the equation expresses C as a function of w.

C(w) = 20w² + () w > 0

Piecewise Defined Functions:

Where the graph is not continuous and is broken into pieces, or has a sharp corner.

Example:

1

1

A function f is defined by:

1-x if x² if x>1

f(x)= {

Since 01, we have f(0)= 1-0=1

Since 11, we have f(1)= 1-1=0

Since 2 >1, we have f(2)= 2² = 4

Absolute Value: The absolute value of a, denoted by , is the distance from a to 0 on the real number line. Distances are always positive or 0 so we have: for every a.

Symmetry:

If a function f satisfies f(-x)= f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x) = x² is even because:

f(-x) = (-x)² = x² = f(x)

If f satisfies f(-x) = -f(x) for every number x in its domain, then f is called an odd function. For example, the function f(x) = x³ is odd because:

f(-x) = (-x)³ = -x³ = - f(x)

Example: Determine whether the following function is even, odd, or neither.

f(x)=x⁵+x

Solution:

f(-x)=(-x) ⁵+(-x)=(-1) ⁵ x⁵+(-x)

=-x ⁵-x=-(x ⁵+x)

=- f(x)

Therefore, f is an odd function

Increasing and Decreasing Functions:

A function f is called increasing on an interval I if

f(x₁) < f(x₂) whenever x₁ < x₂ in I

A function f is called decreasing on an interval I if

f(x₁) > f(x₂) whenever x₁ < x₂ in I

For Extra Help, Visit these resources:

http://math.usask.ca/maclean/110/00/Printables/BW/FourWays.pdf

http://www.blc.edu/fac/rbuelow/calc/nt1-1.htm