Definition of Functions: Do Now: Let If a variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say y is a function of x. i.e. y changes its behavior as x changes its value. ***If an x-value yields more than one y-value, then the equation is NOT a function. Common Notations of Functions: Note: These notation replace the traditional y. For example, instead of: we use: IS ?

This formula is used to find the slope of a secant line. DIFFERENCE QUOTIENT: One of the basic definitions in calculus employs the following ratio This formula is used to find the slope of a secant line. secant line Ex) Evaluate the difference quotient of First evaluate

Evaluate the difference quotient and simplify:

A right triangle has a hypotenuse that is 7 inches long A right triangle has a hypotenuse that is 7 inches long. Express the length of one of the legs of the triangle as a function of the length of the other leg. Express the radius of a circle as a function of the area of the circle. a b 7 This is pronounced “a of b”

DOMAIN OF FUNCTIONS The domain of a function is the set of all x-values for which the function is defined; for which the formula yields a real value. We must be cautious about the following situations when stating a domain: DIVISION BY ZERO NEGATIVE UNDER A RADICAL State the domain of each of the following functions: All real values except x = 0 All real values All real values except x = -3

DO NOW DO NOT USE THE CALCULATOR Is the following function positive or negative at x = 1.9243? Think of a clever way to figure this out that requires little work!

-3 4 Do Now: (DO NOT USE THE CALCULATOR) Solve algebraically. Solve for the critical values and set up the test intervals and the chart to determine which intervals are the solutions to this inequality. TEST INTERVALS SHOULD NOT INCLUDE CRITICAL POINTS; USE PARENTHESIS TEST INTERVALS PICK x- value POLYNONIAL VALUE CONLC. -3 4 x = -2 is an extraneous root (ASSYMTOTE). Therefore, we must use a parenthesis to show that -2 is NOT included in the interval, even though the original inequality has a greater than or equal to symbol, which would have otherwise required ‘]’ Answer:

What if you are asked to state the domain of a function that can be reduced? Do you state the domain before or after simplification? OR The domain of the function after reducing is all real numbers; the algebraic simplification has altered the domain of the function. The domain of the function before reducing is all real numbers except x = 2 The geometric effect of the algebraic cancellation is to eliminate the hole in the original graph. The graph of this equation is said to have a HOLE at x = 2.

What is the difference between a HOLE and an ASSYMTOTE? The function may be continuous before and after the hole. The function approaches the asymptote infinitely but will never cross or go beyond it.

Set up an equation or inequality to find the domain Set up an equation or inequality to find the domain. Write answer in interval notation. All real value except x = 1 All real value except x = 0 and x = -5 Solve as a rational inequality