Functions. 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.

Functions

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.

FUNCTION

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

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.

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

13

2y x

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

22 1x y

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)

3For the function defined by 3 2 , evaluate: f f x x x

Find the domain of a function

Worry in what 3 cases?

Watch for and Know the Three DOMAIN issues…

1. Dividing by zero

2. Even roots of negatives

3. Logs of non-positives

2

4(a)

2 3

xf x

x x

2(b) 9g x x

(c) 3 2h x x

Example

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, 3), (3, 3), (2, -1)}

{(2, 3), (4, 1), (3, -2), (2, -1)}

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.

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

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.

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}

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

YOU CAN DRAW AN INVERSE USING YOUR CALCULATOR IF THE FUNCTION HAS BEEN GRAPHED

FINDING THE INVERSE OF A ONE-TO-ONE FUNCTION

1.Rewrite f(x) as y2.Switch x and y3.Solve for y4.Rewrite y as f-1(x)5.Verify f(f-1(x))= f-

1(f(x))=x