Download Calculating the Domain and the Range of Functions

Domain
What are all the possible values of x?
Are there any x values for the function that the
function is undefined?
Start off with the assumption that the domain is
from -∞ to +∞.
Then, check the function to see if there are any
points where the function is undefined.
Line
Parabola
Ask Yourself: “What can’t x be?”
-∞
+∞
Interval notation
(-∞,+∞)
Interval notation
(-∞,+∞)
Set Notation
{x| all real numbers}
Set Notation
{x| all real numbers}
If n is even, everything
under the radical sign
needs to be positive
( p 0)
If n is odd, there are no restrictions
(Domain and Range would be all real
numbers).
Things that restrict Domain:
Radicals:
n
p
Example:
4
x 5
x 5 0
x
5
Domain: [-5, ∞)
Rational Functions: y
Px
Qx
You can’t divide by zero. Any zero of Q(x) is not included in the
domain. Set Q(x) equal to zero and solve for x. This will give you
values x cannot be.
Example:
x
x 1
x 1 0
x 1
Domain: (-∞, 1) U (1, ∞)
Range
What are all the possible values of y?
Are there any solutions for the function f(x)
that don’t exist?
Ask Yourself: “What can’t y be?”
Start off with the assumption that the range is
from -∞ to +∞.
Then, check the function to see if there are any
points where the function is undefined.
+∞
-∞
Line
Parabola
2
Equation 1: y f x a x h
k
a. Vertex is at (h, k)
Equation 2: y f x ax 2 bx c
b. Vertex is at
b
,c
2a
2
b
4a
Interval notation
(-∞,+∞)
Interval notation
Using equation 1:
If a is positive
(a>0) then:
Range: *k, ∞)
If a is negative
(a<0) then:
Range: (-∞, k+
Using equation 2:
If a>0 then:
Range:
c
b2
,
4a
{y| y≥ c
b2
}
4a
If a<0, then Range is:
{y| y≤k}
or
b2
{y| y≤ c
}
4a
Why does it work like that?
Remember, if a>0 that means we have a parabola
that opens upwards like so:
If a<0 then:
Range:
,c
Set Notation
{y| all real numbers}
Set Notation
if a>0, then Range is:
{y| y≥k}
or
b2
4a
Example (1):
y
2x 3
Range: 1,
2
1
Example (2):
y
x2
2x 1
Range:
22
,1
4( 1)
,1
,2
4
4
Notice how the parabola doesn’t exist below the
vertex (0,0)? That means there are no y values
below the lowest point (the vertex), hence the
range needs to show that.
So, if you flip the graph upside down, then a will
be negative (a<0). You are going to get a parabola
who’s vertex is the maximum y can be; hence the
Range would go from negative infinity to the yvalue of the vertex.
Things that restrict Range:
Radicals:
n
If n is even, everything
under the radical sign
needs to be positive
( p 0 ) , that means y can
never be negative (unless
there is a negative in front
of the radical).
p
If n is odd, there are no restrictions
(Domain and Range would be all real
numbers).
Example (1):
y
x 2
Range: *0,∞) or y
0
Example (2):
y
x 4
Range: (-∞,0+ or y
Rational Functions: y
n
or y
Qx
(where n is any whole, real number)
Px
Qx
0
This is where things get interesting. If the numerator of a rational
function is a number (n) and the bottom contains a function with
respect to x (Q(x)) then the range is all real numbers but zero. Why?
Well, let me show you….
Example (1):
y
2
x 1
Because there is no value of x that will give me an answer (or “y
equals”) zero, then y can never be zero. Hence,
Range: (-∞,0) U (0, ∞) or y≠0
Now, with that in mind, if you have the following:
Example (2):
y
1
3
x 2
The term
1
x 2
can’t be zero, because of that it means that there is
no possible way of the right hand side adding up to 3. Hence,
Range: (-∞,3) U (3, ∞) or y≠3
Now, if the numerator were a function P(x) what happens? Right now,
let’s just worry about if the numerator and denominator are of the
same order (the highest exponent of the numerator is the same as
the highest exponent of the denominator)—when working with
asymptotes you will learn what happens if the order of the numerator
is higher or lower than the denominator. What happens is the Range
is always (-∞,1) U (1, ∞)
Example (3):
y
x
x 3
With long division:
3
x
x x 3
1
Where the quotient is: 1
The
3
x
3
can never be zero that means the sum can never be 1. Hence,
x
Range: (-∞,1) U (1, ∞) or y≠1