1. Functions

2. Definitions Relation: the correspondence between 2 sets
Domain: The set X Range: The set Y
Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element of Y.

3. FUNCTION

5. Is it a function It must have only one y for any given x.
1. Solve the equation for y.
2. Use your prior knowledge…
x2 + y2 = 1

6. Graphs of functions
3. If provided with a graph, we can determine if it is a function by the VERTICAL LINE TEST.
Vertical Line Test: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.

7. Determine if the equation defines y as a function of x.

8. Function Notation y = f(x) so instead of saying y = 2x + 3
we say: f(x) = 2x + 3
Input or Domain is x (independent)
Output or Range is f(x) (dependent)

10. Find the domain of a function
Worry in what 3 cases?

11. Watch for and Know the Three DOMAIN issues…
Dividing by zero
Even roots of negatives
Logs of non-positives

13. Example

15. Determine whether each relation represents a function
Determine whether each relation represents a function. If it is a function, state the domain and range. State the inverse, determine if it is a function and whether the relation is one-to-one.
{(2, 3), (4, 1), (3, -2), (2, -1)}
{(-2, 3), (4, 1), (3, -2), (2, -1)}
{(2, 3), (4, 3), (3, 3), (2, -1)}

16. This function is not one-to-one because two different inputs, 55 and 62, have the same output of 38.
This function is one-to-one because there are no two distinct inputs that correspond to the same output.

17. For each function, use the graph to determine whether the function is one-to-one.

18. A function that is increasing on an interval I is a one-to-one function in I.
A function that is decreasing on an interval I is a one-to-one function on I.

19. State the domain and range of the function and its inverse.
Find the inverse of the following one-to-one function:
{(-5,1),(3,3),(0,0), (2,-4), (7, -8)}
State the domain and range of the function and its inverse.
The inverse is found by interchanging the entries in each ordered pair:
{(1,-5),(3,3),(0,0), (-4,2), (-8,7)}
The domain of the function is {-5, 0, 2, 3, 7}
The range of the function is {-8, -4,0 ,1, 3). This is also the domain of the inverse function.
The range of the inverse function is {-5, 0, 2, 3, 7}

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23. YOU CAN DRAW AN INVERSE USING YOUR CALCULATOR IF THE FUNCTION HAS BEEN GRAPHED
Copyright © 2013 Pearson Education, Inc. All rights reserved

25. FINDING THE INVERSE OF A ONE-TO-ONE FUNCTION
Rewrite f(x) as y
Switch x and y
Solve for y
Rewrite y as f-1(x)
Verify f(f-1(x))= f-1(f(x))=x