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# Functions.

## Presentation on theme: "Functions."— Presentation transcript:

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.

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)

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

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

Example

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)}

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.

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}

Copyright © 2013 Pearson Education, Inc. All rights reserved

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

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

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