Business Math

1006: Section 3.1 Functions

Definition of Function

• A function consists of a set of inputs called the domain, a set of outputs called the range, and a rule by which each input determines exactly one output.

• In other words, only one y for every x. (x can NOT be repeated)

• Domain = independent variable

• Range = dependent variable

• Independent DEPENDS on Dependent

EXAMPLES

• Example 1: The amount of income tax you pay depends on the amount of your income. The way in which the income determines the tax is given by the tax law.– Identify domain and range.

• Example 2: Which of the following describes a function?– Use the optical reader at the checkout counter of the

supermarket to convert codes to prices.– Enter a number in a calculator and press the x2 key.– Assign to each number c the number y given by the

equation y = 3x – 5.– Assign to each number x the number y given by this

table.X 1 1 2 2 3 3

Y 3 -3 -5 -5 8 -8

Domain• Unless otherwise stated, assume that the

domain of any function defined by a formula or an equation is the largest set of real numbers (inputs) that each produce a real number as output.

• Example 3: Find the domain of each function.

• Y=x4

• Y=√(x - 6)• Y=√(4 - x)• Y=1/(x + 3)• Y=√(x)/(x2 - 3x + 2)

Functional Notation

• f(x) is read “f of x” where f is the function name and x is the independent variable that the equation is written in terms of.

• f(3) means I replace x with 3 anywhere there is an x in the equation. The same is done with f(a + b).– NOTICE: f(a +b) ≠ f(a) + f(b)

Examples

• Example 4: Let g(x) = -x2 + 4x – 5. Evaluate the following:

• g(-2)• g(x + h)

• g(x + h) – g(x)

• (g(x + h) – g(x))/h

Applications• The quotient found in Example 4(d) is called the difference quotient of g. These are

important in calculus.• Example 5 (Finance): If you were a single person in Connecticut in 2008 with a taxable

income of x dollars, then your state income tax T was determined by the rule

T(x) = .03x if 0 ≤ x ≤ 10,000

300 + .05(x – 10,000) if x > 10,000

Find the income tax paid by a single person with the given taxable income.

(a) $9200

(b) $30,000

*A function with a multipart rule is called a piecewise-defined function.• Example 6 (Business): Suppose the projected sales (in thousands of dollars) of a small

company over the next 10 years are approximated by the function

S(x) = .07x4 - .05x3 + 2x2 + 7x + 62

(a) What are the projected sales for the current year?

(b) What will sales be in 4 years?