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Learning Objectives for Section 2.1 Functions

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 :

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Learning Objectives for Section 2.1 Functions

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  1. Learning Objectives for Section 2.1Functions • 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.

  2. 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 ___________________.

  3. 10 9.00 12 10.00 16 12.00 domain D range R 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.

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

  5. Examples Determine if the following relations are functions?

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

  7. 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).

  8. Examples These are functions: These are NOT functions:

  9. 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.

  10. 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.

  11. 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

  12. Evaluating Functions • Consider our function • What does f (-3) mean?

  13. Some Examples • Given , complete the following.

  14. Domain of a Function • Consider • Question: For what values of x is the function defined?

  15. Domain of a Function(continued) • Answer:To find the domain of To find the domain of a square root function: Set the radicand (expression under the radical) ≥ 0

  16. Domain of a Function(continued) • Example: Find the domain of the function

  17. Domain of a Function:A Rational Example • Find the domain of

  18. 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 equalto zero to determine which x values must be EXCLUDED. • The domain will be all real numbers EXCEPT for these values.

  19. Domain of a Function:Another Example • Find the domain of

  20. Domain of a Function:And Another Example • Find the domain of

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

  22. 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.

  23. Range of a Function Example: State the range of g using interval notation.

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