Learning Objectives
Introduction
In this tutorial we will be taking a look at working with functions. We will start off by looking for the domain and range. Then, we will move on to finding functional values. Don't be thrown by the different way function notation looks. When you go to evaluate these functions, we are just plugging in values for x and simplifying, just like you have done a million times before (it is one of those types of problems that I warn students to not make harder than it is). Just think of finding a functional value as a fancy assignment statement. I think we are ready to forge ahead.
Tutorial
Be careful. Not every relation is a function. A function has to fit the above definition to a tee.
If you got {3, 4, 5, 6}, 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 {2, 3, 4, 5}, you are absolutely right!
Is this a
function or not?
We need to ask ourselves, does every
first element
(or input) correspond with EXACTLY ONE second element (or
output)?
In this case, the answer is yes. 3 only goes with 2, 4 only goes
with 3, 5 only goes with 4 and 6 only goes with 5.
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 {2, 3, 4}, 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 {3, 4}, you are absolutely right!
Again, note that if any value repeats, we only need to
list it one time.
Is this a
function or not?
We need to ask ourselves, does every first element (or input)
correspond
with exactly one second element (or output)? In this case, the
answer
is NO. The input value of 2 goes with two output values, 3 and
4.
It only takes one input value to associate with more than one output
value
to be invalid as a function.
In this case, the relation is not 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 {a, b, c, d},
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 {1, 2}, you are absolutely right!
Is this a
function or not?
We need to ask ourselves, does every
first element
(or input) correspond with exactly one second element (or
output)?
In this case, the answer is yes. a only
goes with 1, b only goes with 1, c only
goes with 2 and d only goes with 2.
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.
*Inverse of mult. by 5 is div.
by 5
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, 2/5.
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 -16, then our output value y could either be 4 or - 4. 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.
Think about it, if a vertical line intersects a graph in more than one place, then the x value (input) would associate with more than one y value (output), and you know what that means. The relation is not a function.
The next two examples illustrate this concept.
Example 6: Find the domain and range of each relation. Use the vertical line test to determine whether each graph is a graph of a function.
Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the line has arrows at both ends, that means it would go on and on forever to the right and to the left.
This means that the domain is {x | x is a real number}.
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.
In terms of this two dimensional graph, that corresponds with the y values
(vertical axis).
Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the line has arrows at both ends, that means it would go on and on forever up and down.
This means that the range is {y | y is a real number}.
Vertical Line
Test
This graph would pass the vertical line test, because there would not
be any place on it that we could draw a vertical line and it would
intersect
it in more than one place.
Therefore, this is a graph of a function.
Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note that the farthest left point is (1, 0) and the farthest right point is (5, 0), and the circle is enclosed between these values. That means that if we wrote out ordered pairs for all the values going around the circle, we would only use the values from 1 to 5 for x.
This means that the domain is {x | 1 <x< 5}.
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.
In terms of this two dimensional graph, that corresponds with the y values
(vertical axis).
Since that is the case, we need to look up and down and
see if
there are any end points. In this case, note that the
highest
point is (3, 2) and the lowest point is (3, -2), and the circle is
enclosed
between these values. That means that if we wrote out ordered
pairs
for all the values going around the circle, we would only use the
values
from -2 to 2 for y.
This means that the range is {y |
-2 < y < 2}
Vertical Line
Test
This graph would not pass the vertical line test because there is at
least one place on it that we could draw a vertical line and intersect
it in more than one place. In fact, there are a lot of vertical
lines
that we can draw that would intersect it in more than one place, but we
only need to show one to say it is not a function.
The graph below shows one vertical line drawn through our graph that intersects it in two places: (3, 2) and (3, -2). This shows that the input value of 3 associates with two output values, which is not acceptable in the function world.
Therefore, this is not a graph of a function.
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. Just like you do when you are finding the value of an expression when you are given a number for your variable as shown in Tutorial 2: Algebraic Expressions. For example, if we are looking for f(0), we would plug in 0 for the value of x in our function f.
*Plug in -1 for x and evaluate
*Plug in 1 for x and evaluate
*Plug in 1/2 for x and evaluate
*Plug in 5 for x and evaluate
Note how there is no input value visible. This is not the same as having 3x.
*This is a constant function
that is always
3
*This is a constant function that is always 3
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 Problem 1a - 1b: Find the domain and range of the relation. Also, determine whether the relation is a function.
Practice Problems 2a - 2b: Find the domain and range of each relation. Use the vertical line test to determine whether each graph is a graph of a function.
2b.
Practice Problems 3a - 3b: Decide whether y is a function of x.
Practice Problems 4a - 4b: Find the functional values.
http://www.purplemath.com/modules/fcns.htm
This website goes over what a function is and what domain and range
are.
http://www.purplemath.com/modules/fcnnot.htm
This website goes over function notation.
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 July 3, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.