Learning Objectives
Introduction
Tutorial
Be careful. Not every relation is a function. A function has to fit the above definition to a tee.
DomainSo, this relation would not be
an example of
a function.
Domain
We need to find the set of all input values. In terms of ordered
pairs, that correlates with the first component of each one. So,
what do you get for the domain?
If you got {1, 2, 3}, you are correct!
Note that if any value repeats, we only need to list it
one time.
Range
We need to find the set of all output values. In terms of ordered
pairs, that correlates with the second component of each one. So,
what do you get for the range?
If you got {2, -2, 3, 4}, you are absolutely right!
Note that a relation can still be a function if an output value associates with more than one input value as shown in this example. But again, it would be a no no the other way around, where an input value corresponds to two or more output values.
So, this relation would be an
example of a
function.
Domain
We need to find the set of all input values. In terms of ordered
pairs, that correlates with the first component of each one. So,
what do you get for the domain?
If you got {5, 10, 15}, you are correct!
Range
We need to find the set of all output values. In terms of ordered
pairs, that correlates with the second component of each one. SO,
what do you get for the range?
If you got {10}, you are absolutely right!
Note that if any value repeats, we only need to list it one time.
*Inverse of mult. by -2y is div. by -2y
*Solved for y
If you answered yes, you are right on. For example, if we plugged in a 1 for x, then y would only equal one value, -1/2. Note that ANY value you would plug in for x would produce only one value for y.
Note that since it is solved for y, y is our output value and x is our input value.
Since our answer to that question is yes, that means by definition, y is a function of x.
If you answered no, you are correct.
For example, if our input value x is 3, then our output value y could either be 2 or -2. Note that I could have picked an infinite number of examples like this one. You only need to show one example where the input value is associated with more than one output value to disqualify it from being a function.
This means that at least one input value is associated with more than one output value, so by definition, y is not a function of x.
Function Notationx is your input variable.
Think of functional notation as a fancy assignment statement. When you need to evaluate the function for a given value of x, you simply replace x with that given value and simplify. For example, if we are looking for f(0), we would plug in 0 as the value of x in our function f.
*Plug in 0 for x and evaluate
*Plug in 1 for x and evaluate
*Plug in -5 for x and evaluate
*Plug in 0 for x and evaluate
*Plug in 5 for x and evaluate
Don't let the fact that we need to plug in an a throw you. You plug it into the function just like you do a number. Everywhere you have an x in your function, replace it with an a.
Don't let the fact that we need to plug in the expression a + h throw you. You plug it into the function just like you do a number. Everywhere you have an x in your function, replace it with an a + h.
Putting it all together we get:
*Combine like terms
*Divide out h from every term
A function of the form
f(x) = C,
where C is a constant.
Example
8: Find the functional values h(0) and h(2) of the constant function h(x)
= -5.
The rule for specifying it is given by more than one expression.
Example
9: Find the functional values f(1), f(3),
and f(4) for the compound function
*Plug in 4 for x
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1b: Determine if the given relation is function or not. Give its domain and range.
Practice Problems 2a - 2b: Decide whether y is a function of x.
Practice Problem 3a: Find f(-5) and f(2) for the given function.
Practice Problem 4a: Find and simplify a) f(a), b) f(a + h) and
using the given function.
Practice Problem 5a: Find the functional values g(-1), g(0), and g(4) for the compound function.
Practice Problems 6a - 6c: Give the domain of the function.
Need Extra Help on these Topics?
The following are webpages that can assist you in the topics that were covered on this page:
http://www.purplemath.com/modules/fcnnot.htm
This website goes over function notation.
http://www.purplemath.com/modules/fcns.htm
This website goes over what a function is and what domain and range
are.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on March 30, 2010 by Kim Seward.
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