Learning Objectives for Section 2.1 Functions MAT 103 SPRING 2009 Learning Objectives for Section 2.1 Functions You will be able to give and apply the definition of a function. You will be able to identify domain and range of a function. You will be able to use function notation.

Function Definition Definition of Function: MAT 103 SPRING 2009 Function Definition Definition of Function: A function is a correspondence between two sets D and R such that each element of the first set, D, maps to EXACTLY one element of the second set, R. The first set (inputs) is called the __________________, and the set of corresponding elements in the second set (outputs) is called the ___________________.

MAT 103 SPRING 2009 Function Diagram You can visualize a function by the following diagram which shows a correspondence between two sets: D, the domain of the function, gives the diameter of pizzas, and R, the range of the function gives the cost of the pizza. 10 12 16 9.00 12.00 10.00 domain D range R

Examples Determine if the following relations are functions. 1) 2) 3 1) 2) 3 -2 8 1 2 3 1 5 6

Examples Determine if the following relations are functions.

Functions Specified by Equations MAT 103 SPRING 2009 Functions Specified by Equations Consider the equation Input: x = -2 -2 Input Output Process: square (–2), then subtract 2 (-2, 2) is an ordered pair of the function. 2 Output: result is 2

Functions as Equations To determine if an equation represents y as a function of x: You should be able to solve for y, using only one equation, so that each input (x) has only one output (y). Note: If y is raised to an even power or if y is in the absolute value, then the equation does NOT represent y as a function of x.

Examples These are functions: These are NOT functions:

Vertical Line Test for a Function MAT 103 SPRING 2009 Vertical Line Test for a Function If you have the graph of an equation, there is an easy way to determine if it is the graph of a function. It is called the ________________ ___________ ___________ An equation defines a function if each vertical line in the coordinate system passes through AT MOST ONE POINT on the graph of the equation. IF ANY VERTICAL LINE PASSES THROUGH TWO OR MORE POINTS ON THE GRAPH, THEN THE EQUATION DOES NOT DEFINE A FUNCTION.

Vertical Line Test for a Function (continued) MAT 103 SPRING 2009 Vertical Line Test for a Function (continued) This graph is not the graph of a function because you can draw a vertical line which crosses it twice. This is the graph of a function because any vertical line crosses only once.

MAT 103 SPRING 2009 Functional Notation FUNCTIONAL NOTATION is used to describe functions. The variable y will now be called f (x). f (x) is read as “ f of x” and simply means the y coordinate of the function corresponding to a given x value. Our previous equation can now be expressed as

Evaluating Functions Consider our function What does f (-3) mean? MAT 103 SPRING 2009 Evaluating Functions Consider our function What does f (-3) mean?

MAT 103 SPRING 2009 Some Examples Given , complete the following.

Domain of a Function Consider MAT 103 SPRING 2009 Domain of a Function Consider Question: For what values of x is the function defined?

Domain of a Function (continued) MAT 103 SPRING 2009 Domain of a Function (continued) To find the domain of a square root function: Set the radicand (expression under the radical) ≥ 0 and solve for x. Answer: To find the domain of

Domain of a Function (continued) MAT 103 SPRING 2009 Domain of a Function (continued) Example: Find the domain of the function

Domain of a Function: A Rational Example MAT 103 SPRING 2009 Domain of a Function: A Rational Example Find the domain of

Domain of a Rational Function MAT 103 SPRING 2009 Domain of a Rational Function Remember, if the denominator is equal to zero, the function is _________________________! To find the domain of a rational function (variable in the denominator): Set the denominator equal to zero and solve for x to determine which x values must be EXCLUDED. The domain will be all real numbers EXCEPT for these values.

Domain of a Function: Another Example MAT 103 SPRING 2009 Domain of a Function: Another Example Find the domain of

Domain of a Function: And Another Example MAT 103 SPRING 2009 Domain of a Function: And Another Example Find the domain of

MAT 103 SPRING 2009 Domain of a Function The domain of any unrestricted linear, quadratic, cubic function, or absolute value function is the set of ____________ __________________. For example, the following functions have a domain of Check these out on your graphing calculator.

MAT 103 SPRING 2009 Range of a Function The range of a function is the set of all outputs (y-values) . To find the range of a function, look at its graph to determine the y-values that are included on the graph. Check if your graph has a maximum or minimum y-value to help you determine the range.

Example: State the range of g using interval notation. MAT 103 SPRING 2009 Range of a Function Example: State the range of g using interval notation.