Domain and range (linear, quadratic, rational functions)

Domain and Range
Linear, Quadratic, and Rational Functions

Remember
Functions
Function: a relation in which, for each value of the first component of
the ordered pairs, there is exactly one value of the second component
•A function is a relation in which the members of the domain
(x-values) DO NOT repeat.
•So, for every x-value there is only one y-value that corresponds
to it.
•y-values CAN be repeated.

Finding Domain & Range
Give the domain & range of each relation. Is it a
function?

Example 5
Domain = { 0 ≤ x < 3}

Example 7 (Quadratic Function)
plug any values
of x into the
function and it
will produce a
valid output. So,
I can safely say
that its domain
is all x values
The graph of the parabola
has a low point at y = 3 and
it can go as high as it wants.
Using inequality, I will write
the range as y ≥ 3.

Example 8 (Quadratic Function)
a quadratic function
will always have a
domain of all x
values.
The parabola opens upward and the vertex
must be a minimum. The coordinate of the
vertex is…
We can now see that this
parabola has a minimum value
at y = −5, and can go up to
positive infinity.
The range should be y ≥ −5.

Example 9 (Irrational / Radical Function)
let the expression
under the radical,
x-2, greater than
or equal to zero;
and then solve the
inequality. Check
out my other
lesson on how to
solve inequalities.
This radical
function has a
domain of x ≥ 2.
The radical function starts at
y = 0 and can go as high as
it wants (positive infinity).
We will claim that the range
of this function is y ≥ 0.

Example 10 (Irrational / Radical Function)
The acceptable
values under the
square root are
zero and positive
numbers.
Now, the domain
of the function is
x ≤ 5
The radical
function starts at y
= 0, and then
slowly but steadily
decreasing in
values all the way
down to negative
infinity. This
makes the range y
≤ 0.

Example 11 (Rational Function)
our domain is all x-values but
does not include x = 2. It makes
a lot of sense because we can plug
any values of x into the function
with the exception of x = 2, and the
function will have valid outputs.
This function contains a
denominator. This tells me
that I must find the x-values
that can make the
denominator zero to
prevent the undefined case
to happen. the range is all
y-values but
does not include
y = 0. The open
circle in the graph
below denotes that
y = 0 is excluded
from the range.

Example 12 (Rational Function)
our domain is all x-values but
does not include x = 2. It makes
a lot of sense because we can plug
any values of x into the function
with the exception of x = 2, and the
function will have valid outputs.
The idea again is to exclude
the values of x that can make
the denominator zero.
Obviously, that value is x = 2
and so the domain is all x
values except x = 2.
the graph shows
that it covers all
possible y-values:
goes up and down
without bounds,
and no breaks in
between.
Therefore the
range is all y
values.

•Consider the following relation:
•
•Is this a function?
•What is domain and range?
Example 13

Visualizing range and domain of

Domain and range (linear, quadratic, rational functions)

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Domain and range (linear, quadratic, rational functions)