Elana owns a consulting business that helps software companies market their services to school districts. She earns an average of $5687.1 for every contract one of her client companies signs with her help. In order to be able to run her business, she needs to cover $7,000/month (rent, licenses, etc.). The average costs associated with each contract are $1260.7. How many contracts must she facilitate each month in order to break even

2 contracts

Step-by-step explanation:

Break even point refers to the number of units or sales that needs to be generated for the company to make neither a profit nor a loss.

This means that at the break even point, sales is equivalent to the cost incurred (both fixed and variable).

Let the number of contracts that must be signed to break even be s

The rent is a fixed cost while the cost associated with each contract is variable.

5687.1s = 7000 + 1260.7s

5687.1s - 1260.7s = 7000

4426.4 s = 7000

s = 1.58

≈ 2

She must facilitate 2 contracts each month to break even.

when x is less than 2, it gives x2,

when x is exactly 2 it gives 6

when x is more than 2 and less than or equal to 6 it gives the line 10-x

It looks like this:

Piecewise Function

(a solid dot means "including",

an open dot means "not including")

And this is how we write it:

Piecewise Function

The Domain (all the values that can go into the function) is all Real Numbers up to and including 6, which we can write like this:

Dom(f) = (-∞, 6] (using Interval Notation)

Dom(f) = {x member of Reals | x ≤ 6} (using Set Builder Notation)

And here are some example values:

XY

−416

−24

00

11

26

37

Example: Here is another piecewise function:

continuous jump fn h(x) = 2 if x<=1, x if x>1

which looks like:  continuous jump graph h(x)

What is h(−1)?

x is ≤ 1, so we use h(x) = 2, so h(−1) = 2

What is h(1)?

x is ≤ 1, so we use h(x) = 2, so h(1) = 2

What is h(4)?

x is > 1, so we use h(x) = x, so h(4) = 4

Piecewise functions let us make functions that do anything we want!

Example: A Doctor's fee is based on the length of time.

Up to 6 minutes costs $50

Over 6 and up to 15 minutes costs $80

Over 15 minutes costs $80 plus $5 per minute above 15 minutes

Which we can write like this:

function piecewise f(t)=50 if t<=6, 80 if t>6 and t<=15, 80 + 5(t-15) if t>15

You visit for 12 minutes, what is the fee? $80

You visit for 20 minutes, what is the fee? $80+$5(20-15) = $105

The Absolute Value Function

The Absolute Value Function is a famous Piecewise Function.

It has two pieces:

below zero: -x

from 0 onwards: x

Absolute Value function

f(x) = |x|

Absolute Value function

The Floor Function

The Floor Function is a very special piecewise function. It has an infinite number of pieces:

Floor function

The Floor Function