Functions:
Introduction:
Functions are a correspondence between two sets "say X and Y, called the domain
and the range. When defining a function, you usually state what kind of numbers the
domain (x) and range (f(x)) values can be. But even if you say they are real numbers,
that doesn’t mean that all real numbers can be used for x. It also doesn’t mean that all
real numbers can be function values, f(x). There may be restrictions on the domain
and range. The restrictions partly depend on the type of function.
In this topic, all functions will be restricted to real number values. That is, only real
numbers can be used in the domain, and only real numbers can be in the range.
Types of functions:
Constant Function:
Let 'A' and 'B' be any two non–empty sets, then a function '
Constant Function if and only if range of '
OR;
Let
if
be a function then a function '
,
whose
' from 'A' to 'B' is called
' is a singleton.
' from 'A' to 'B' is called a constant function
is a constant.
e.g.
Algebraic Function:
The function defined by algebraic expression are called algebraic function.
e.g.
Polynomial Function:
A function of the form
Where 'n' is a positive integer and
are real number is called a polynomial
function of degree 'n'.
Linear Function:
A polynomial function with degree ' ' is called a linear function. The most general form of
linear function is
Quadratic Function:
A polynomial function with degree '2' is called a Quadratic function.
The most general form of Quadratic equation is
Cubic Function:
A polynomial function with degree '3' is called cubic function.
The most general form of cubic function is
Identity Function:
Let
be a function then '
' is called on identity function. If
.
Rational Function:
A function
defined by
function is called, rational function.
, where both
and
are polynomial
Trigonometric Function:
A function
,
etc, then
is called trigonometric function.
Exponential Function:
A function in which the variable appears as exponent (power) is called an exponential
function
e.g. (i)
(ii)
.
Logarithmic Function:
A function in which the variable appears as an argument of logarithmic is called
logarithmic function.
e.g.
.
Restricting the domain
There are two main reasons why domains are restricted.
 You can’t divide by 0.
 You can’t take the square (or other even) root of a negative number,
as the result will not be a real number.
In what kind of functions would these two issues occur?
Division by 0 could happen whenever the function has a variable in the denominator
of a rational expression. That is, it’s something to look for in rational functions. Look
at these examples, and note that “division by 0” doesn’t necessarily mean that x is 0!
Function
Notes
If x = 0, you would be dividing by 0, so x ≠ 0.
If x = 3, you would be dividing by 0, so x ≠ 3.
Although you can simplify this function to
f(x) = 2, when x = 1 the original function would
include division by 0. So x ≠ 1.
Both x = 1 and x = −1 would make the
denominator 0. Again, this function can be
simplified to
, but when x = 1 or x =
−1 the original function would include division by
0, so x ≠ 1 and x ≠ −1.
This is an example with no domain restrictions,
even though there is a variable in the
denominator. Since x2 ≥ 0, x2 + 1 can never be
0. The least it can be is 1, so there is no danger
of division by 0.
Square roots of negative numbers could happen whenever the function has a
variable under a radical with an even root. Look at these examples, and note that
“square root of a negative variable” doesn’t necessarily mean that the value under
the radical sign is negative! For example, if x = −4, then −x = −(−4) = 4, a positive
number.
Function
Restrictions to the Domain
If x < 0, you would be taking the square root of a
negative number, so x ≥ 0.
If x < −10, you would be taking the square root
of a negative number, so x ≥ −10.
When is -x negative? Only when x is positive.
(For example, if x = −3, then −x = 3. If x = 1,
then −x = −1.) This means x ≤ 0.
x2 – 1 must be positive, x2 – 1 > 0.
So x2 > 1. This happens only when x is greater
than 1 or less than −1: x ≤ −1 or x ≥ 1.
There are no domain restrictions, even though
there is a variable under the radical. Since
x2 ≥ 0, x2 + 10 can never be negative. The least
it can be is 10, so there is no danger of taking
the square root of a negative number.
Domains can be restricted if:


the function is a rational function and the denominator is 0 for some value or
values of x.
the function is a radical function with an even index (such as a square root), and
the radicand can be negative for some value or values of x.
Range
Remember, here the range is restricted to all real numbers. The range is also
determined by the function and the domain. Consider these graphs, and think about
what values of y are possible, and what values (if any) are not. In each case, the
functions are real-valued—that is, x and f(x) can only be real numbers.
Quadratic function, f(x) = x2 – 2x – 3
Remember the basic quadratic function: f(x) = x2 must always be positive, so f(x) ≥ 0
in that case. In general, quadratic functions always have a point with a maximum or
greatest value (if it opens down) or a minimum or least value (it if opens up, like the
one above). That means the range of a quadratic function will always be restricted to
being above the minimum value or below the maximum value. For the function
above, the range is f(x) ≥ −4.
Other polynomial functions with even degrees will have similar range restrictions.
Polynomial functions with odd degrees, like f(x) = x3, will not have restrictions.
Radical function, f(x) =
Square root functions look like half of a parabola, turned on its side. The fact that the
square root portion must always be positive restricts the range of the basic function,
, to only positive values. Changes to that function, such as the negative in
front of the radical or the subtraction of 2, can change the range. The range of the
function above is f(x) ≤ −2.
Rational function, f(x) =
Rational functions may seem tricky. There is nothing in the function that obviously
restricts the range. However, rational functions have asymptotes—lines that the
graph will get close to, but never cross or even touch. As you can see in the graph
above, the domain restriction provides one asymptote, x = 6. The other is the line y =
1, which provides a restriction to the range. In this case, there are no values of x for
which f(x) = 1. So, the range for this function is all real numbers except 1.
Domain and Range:
Finding domain and range of different functions is often a matter of asking yourself,
what values can this function not have?
Example
Problem
What are the domain and range of the real-valued
function f(x) = x + 3?
This is a linear function. Remember that linear functions are lines
that continue forever in each direction.
Any real number can be substituted for x and get a meaningful
output. For any real number, you can always find an x value that
gives you that number for the output. Unless a linear function is a
constant, such as f(x) = 2, there is no restriction on the range.
Answer
The domain and range are all real numbers.
Example
Problem
What are the domain and range of the real-valued
function f(x) = −3x2 + 6x + 1?
This is a quadratic function. There are no rational or radical expressions,
so there is nothing that will restrict the domain. Any real number can be
used for x to get a meaningful output.
Because the coefficient of x2 is negative, it will open downward. With
quadratic functions, remember that there is either a maximum (greatest)
value, or a minimum (least) value. In this case, there is a maximum
value.
The vertex, or turning point, is at (1, 4). From the graph, you can see
that f(x) ≤ 4.
Answer
The domain is all real numbers, and the range is all real
numbers f(x) such that f(x) ≤ 4.
You can check that the vertex is indeed at (1, 4). Since a quadratic function has two
mirror image halves, the line of reflection has to be in the middle of two points with
the same y value. The vertex must lie on the line of reflection, because it’s the only
point that does not have a mirror image!
In the previous example, notice that when x = 2 and when x = 0, the function value is
1. (You can verify this by evaluating f(2) and f(0).) That is, both (2, 1) and (0, 1) are
on the graph. The line of reflection here is x = 1, so the vertex must be at the point (1,
f(1)). Evaluating f(1) gives f(1) = 4, so the vertex is at (1, 4).
Example
Problem
What are the domain and range of the real-valued
?
function
This is a radical function. The domain of a radical function is any x value
for which the radicand (the value under the radical sign) is not negative.
That means x + 5 ≥ 0, so x ≥ −5.
Since the square root must always be positive or 0,
means
.
. That
Answer
The domain is all real numbers x where x ≥ −5, and the
range is all real numbers f(x) such that f(x) ≥ −2.
Example
Problem
What are the domain and range of the real-valued
function
?
This is a rational function. The domain of a rational function is restricted
where the denominator is 0. In this case, x + 2 is the denominator, and
this is 0 only when x = −2.
For the range, create a graph using a graphing utility and look for
asymptotes:
One asymptote, a vertical asymptote, is at x =−2, as you should expect
from the domain restriction. The other, a horizontal asymptote, appears
to be around y = 3. (In fact, it is indeed y = 3.)
Answer
The domain is all real numbers except −2, and the range is
all real numbers except 3.
You can check the horizontal asymptote, y = 3. Is it possible for
3? Write an equation and try to solve it.
to be equal to
Since the attempt to solve ends with a false statement—0 cannot be equal to 6!—the
equation has no solution. There is no value of x for which
that the range is restricted.
, so this proves
Find the domain and range of the real-valued function f(x) = x2 + 7.
A) The domain is all real numbers and the range is all real numbers f(x) such that
f(x) ≥ 7.
B) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers
f(x) such that f(x) ≥ 7.
C) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers.
D) The domain and range are all real numbers.
Answers:
A) The domain is all real numbers and the range is all real numbers f(x) such that
f(x) ≥ 7.
Correct. Quadratic functions have no domain restrictions. Since x2 ≥ 0, x2 + 7 ≥ 7.
B) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers
f(x) such that f(x) ≥ 7.
Incorrect. Negative values can be used for x. The correct answer is: The domain is all
real numbers and the range is all real numbers f(x) such that f(x) ≥ 7.
C) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers.
Incorrect. Negative values can be used for x, but the range is restricted because x2 ≥
0. The correct answer is: The domain is all real numbers and the range is all real
numbers f(x) such that f(x) ≥ 7.
D) The domain and range are all real numbers.
Incorrect. While it’s true that quadratic functions have no domain restrictions, the
range is restricted because x2 ≥ 0. The correct answer is: The domain is all real
numbers and the range is all real numbers f(x) such that f(x) ≥ 7.
Graphing domain and Range:
The domain is the set of all first elements of ordered pairs (x-coordinates).
The range is the set of all second elements of ordered pairs (y-coordinates).
Domain and range can be seen clearly from a graph.
Example 1:
Domain: {1, 3, 4, 6}
Range: {-2, 2, 5}
Example 2:
Domain:
(all real numbers)
Range:
The two examples shown above are functions. But, as we know, not all graphs are
functions.
The graph at the left is:
Since the graph FAILS the Vertical Line
Test, this relation is not a function.
If we restrict the graph to only the "positive"
(or we could have chosen negative) y-values,
the graph will be a function:
(graph
below)
In a similar fashion, we can also restrict domains to ensure that graphs are functions.
The graph at the left is:
If the domain for this graph is listed as "all
Real numbers", this relation is NOT a
function. At first glance this graph appears
to pass the Vertical Line Test, but it is
actually undefined at x = -1.
If we restrict the domain to be "all Real
numbers excluding -1", our relation will be a
function.
Domain: