*y*-values. We have been doing this to find points on the graph (

*x*,

*y*). This process can be streamlined using function notation.

This new notation above is read, "

*y*is equal to

*f*of

*x*". A function is a rule that uniquely assigns one output to every one input. So, at this point we can think of our non-vertical lines as functions.

The

**of a function is the set of inputs, usually the**

*domain**x*-values. The

**of a function is the set of outputs, usually the**

*range**y*-values.

**Evaluate the given function**.

**Find**.

*x*Do not let the function notation discourage you, it takes some time to get used to. The main thing to remember is that

*y*=

*f*(

*x*); somtimes

*x*is given and sometimes

*y*or

*f*(

*x*) is given.

**Graph the given linear function**.

**Word Problem**. Bill has a popular software company which sells copies of its program for $149. If the initial start up cost for the company was $10,000 and it costs $12 to produce each copy:

a. Find a cost function

*C*(

*x*) that models this business.

b. Find a revenue function

*R*(

*x*) that models this business.

c. Find the profit function

*P*(

*x*) using your functions above.

d. Find the profit when 1000 programs are produced and sold.

e. Find the number of programs that must be sold to break even.

**Solution**:

a.

**Cost Function**- Include all fixed and variable costs of production.

b.

**Revenue Function**- Include all proceeds from sales.

c.

**Profit Function**- Revenue less cost of production.

d. Here

*x*= 1,000 programs produced and sold.

e.

**Break Even**- Occurs when profit is equal to zero.

**Video Examples on YouTube**: